Edited By Team Careers360 | Updated on May 07, 2022 12:48 PM IST
One of the chief features in the behaviour of functions is the property known as continuity. For instance, the continuous expansion of a rod on heating, of the continuous growth of an organism, of a continuous flow, or a continuous variation of atmospheric temperature, etc. The idea of continuity of a function stems from the geometric notion of 'no breaks in a graph'. The limit used to define the slope of a tangent line or the instantaneous velocity of a freely falling body is also used to define one of the two fundamental operations of calculus – differentiation.
This article is about the concept of class 12 maths continuity and differentiability. Continuity and differentiability chapter is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE, BCECE, and more.
Continuity and Differentiability
Continuity and differentiability is one of the fundamental concepts of calculus. They help analyze changes, optimize processes, and predict trends in fields like engineering, physics, and economics. Continuity and differentiability allow us to study functions near critical points, even if the function is not defined at those points.
Continuity
Suppose is a real function on a subset of the real numbers and let be a point in the domain of . Then is continuous at if
More elaborately, if the left-hand limit, right-hand limit, and the value of the function at exist and are equal to each other, then is said to be continuous at .
If the right-hand and left-hand limits at coincide, then we say that the common value is the limit of the function at . Hence we may also rephrase the definition of continuity as follows:
A function is continuous at if the function is defined at and if the value of the function at equals the limit of the function at .
If is not continuous at , we say is discontinuous at , and is called a point of discontinuity of .
If the function is continuous, Its graph does not break but for discontinuous functions, there is a break in the graph. A real function is continuous at a fixed point if we can draw the graph of the function around that point without lifting the pen from the plane of the paper. In case one has to lift the pen at a point, the graph of the function is said to have a break or discontinuity at that point, say .
Continuity can be defined in two ways: Continuity at a point and Continuity over an interval.
Continuity at a point
Let us see different types of conditions to see continuity at point
We see that the graph of has a hole at , which means that is undefined. At the very least, for to be continuous at , we need the following conditions:
(i) is defined
Next, for the graph given below, although is defined, the function has a gap at . In this graph, the gap exists because lim does not exist. We must add another condition for continuity at , which is
(ii) exists
The above two conditions by themselves do not guarantee continuity at a point. The function in the figure given below satisfies both of our first two conditions but is still not continuous at . We must add a third condition to our list:
(iii)
So, a function is continuous at a point if and only if the following three conditions are satisfied:
i) is defined
ii) exists
iii) or
A function is discontinuous at a point if it fails to be continuous at .
Continuity over an Interval
Over an open interval : A function is continuous over an open interval if is continuous at every point in the interval.
For any is continuous if
Over a closed interval : A function is continuous over a closed interval of the form if - it is continuous at every point in and - is right-continuous at and - is left-continuous at . i.e.At , we need to check R.H.L. .
L.H.L. should not be evaluated to check continuity of the first element of the interval,
Similarly, at , we need to check L.H.L. .
R.H.L. should not be evaluated to check continuity of the last element of the interval
Consider one example,
, prove that this function is not continuous in ,
Solution:
Condition 1: For continuity in At any point lying in , as lies in LHL at (as in close left neighbourhood of , the function equals 2)
RHL at (as in close right neighbourhood of , the function equals 2)
So function is continuous for any c lying in . Hence the function is continuous in
Condition 2: Right continuity at
So is left continuous at
Condition 3: Left continuity at and
(as in left neghbourhood of ) So does not equal LHL at hence is not left continuous at
So the third condition is not satisfied and hence is not continuous in
Properties of Continuous function
1. If are two continuous functions at a point a of their common domain D. Then fg are continuous at a and if then is also continuous at . Suppose and g be two real functions continuous at a real number c . Then
(1) is continuous at . (2) is continuous at . (3) is continuous at . (4) is continuous at , (provided ). The sum, difference, product, and quotient of two continuous functions are always a continuous function. However is continous function at only if
2. If f is continuous at a and then there exists an open interval ) such that for all has the same sign as
3. If a function is continuous on a closed interval , then it is bounded on ( ) and there exists real numbers k and K such that for all
Differentiability
The instantaneous rate of change of a function with respect to the independent variable is called derivative. Let be a given function of one variable and denotes a number ( positive or negative) to be added to the number .
Let denotes the corresponding change of f , then
If approaches a limit as approaches to zero, this limit is the derivative of at the point . The derivative of a function is denoted by symbols such as or .
The derivative of a function at point is defined by
If does not exists, we say that the function is not differentiable at .
Or, we can say that a function is differentiable at a point ' ' in its domain if limit of the function exists at .
i.e. Right hand limit = RHD= Left hand limit = LHD=
(Both the left-hand derivative and the right-hand derivative are finite and equal.)
is the condition for differentiability at
Examining the Differentiability
1. Using Differentiation (only for continuous functions)
Some functions are defined piecewise, in such cases first we need to check if the function is continuous at the split point, and if it is continuous we need to differentiate each branch function and compare left-hand and right-hand derivative at the split point.
First check if is continuous at . If it is not continuous, then it cannot be differentiable. If it is continuous, then to check differentiability, find
Differentiability can be checked at by comparing
2. Differentiability using Graphs
A function is not differentiable at if 1. The function is discontinuous at 2. The graph of a function has a sharp turn at 3. A function has a vertical tangent at
Illustration 1
Check the differentiability of the following function. 1.
Method 1
Using graphical transformation, we can draw its graph
Using the graph we can tell that at , the graph has a sharp turn, so it is not differentiable at .
Method 2
As , so the function is continuous at So we can use differentiation to check differentiability
As these are not equal, so, is not differentiable at
Illustration 2
Plot the graph of | log | using graphical transformation
We can see that graph has a sharp turn at +1 and -1 so the function is not differentiable at these points.
Properties of Differentiability
1. A function is differentiable in an open in interval ( ) if it is differentiable at every point on the open interval .
2. If is differentiable at every point on the open interval (a,b). And, It is differentiable from the right at " " and the left at " ".(In other words, and both exists), then is said to be differentiable in
3. If a function is differentiable at every point in an interval, then it must be continuous in that interval. But the converse may or may not be true.
Continuity and Differentiability Formulas
Continuity and differentiability class 12 formulas include algebraic properties of continuity and discontinuity, algebraic properties of differentiability, differentiation of implicit functions, and differentiation of functions in parametric form.
If is continuous and is discontinuous, then is a discontinuous function.
Let , which is continuous at and (greatest integer function) which is discontinuous at , are to function.
Now, (fractional part of ) discontinuous at
If is continuous and is discontinuous at then the product of the functions, is may or may not be continuous at .
For example, Consider the functions, . And . is continuous at and is discontinuous at Now,
is continuous at Take another example, consider and is continuous at and is discontinuous at Now,
And we know that signum function is discontinuous at .
If and , both are discontinuous at then the the function obtained by algebraic operation of and may or may not be continuous at .
Rules of Differentiation
Let and be differentiable functions and be a constant. Then each of the following rules of differentiation holds.
Sum Rule
The derivative of the sum of a function and a function is the same as the sum of the derivative of and the derivative of .
In general,
Difference Rule
The derivative of the difference of a function and function is the same as the difference of the derivative of and the derivative of .
Constant Multiple Rule
The derivative of a constant multiplied by a function is the same as the constant multiplied by the derivative of
Product rule
Let and be differentiable functions. Then,
This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
Extending the Product Rule
If three functions are involved, i.e let Let us have a function as the product of the function and . That is, . Thus,
[By applying the product rule to the product of and .]
Quotient Rule
Let and be differentiable functions. Then
OR if , then As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives.
Chain Rule
If and are differentiable funcitons, then or If , then
is known as the chain rule. Or,
The chain rule can be extended as follows If , then
Algebra of Differentiability
The algebraic properties of differentiablity is
1. If and are both differentiable functions at , then the following functions are also differentiable at . (i) (ii) (iii) , provided
2. If is differentiable at and is not differentiable at , then will not be differentiable at .
For example, is not differentiable at , as is differentiable at , but is not differentiable at .
In other cases, and may or may not be differentiable at = a, and hence should be checked using LHD-RHD, continuity or graph.
For example, if (differentiable at ) and (nondifferentiable at ). Their product is which is differentiable. But if , then is non-differentiable at .
3. If and both are nondifferentiable functions at , then the function obtained by the algebraic operation of and may or may not be differentiable at . Hence they should be checked.
For example, Let , not differentiable at , and which is also not differentiable at . Their sum is differentiable and the difference is not differentiable. So there is no definite rule.
4. Differentiation of a continuous function may or may not be continuous.
Differentiation of Implicit Functions
An implicit function is a function that includes both dependent and independent variables such as . For example, the equation of a circle is an implicit function because is not explicitly expressed as a function of . Implicit differentiation means differentiation on both sides of the equation concerning the independent variable with the help of the Chain rule.
To find in such a case, we differentiate both sides of the given relation concerning keeping in mind that the derivative of concerning is
For example
It should be noted that but
Differentiation of Functions in Parametric Form
Parametric differentiation is the process of finding the derivative of the equation in which the dependent variable and independent variable are equated to another variable . To find the derivative of for we use the chain rule.
Sometimes, and are given as functions of a single variable, i.e., and are two functions and is a variable. In such cases, and are called parametric functions or parametric equations and is called the parameter.
To find in such cases, first find the relationship between and by eliminating the parameter and then differentiate concerning .
We have and , in this case, we will differentiate both functions separately. We will first differentiate and concerning ' separately. On differentiating for ' ' we get and on differentiating by ' ' we get .
But sometimes it is not possible to eliminate , then in that case use
For example
If and , then is Solution.
Using the circle example, we differentiate concerning
Importance of Class 12 Continuity and Differentiability
Continuity and differentiability have a significant weighting in the IIT JEE test, which is a national level exam for 12th grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.
How to Study Class 12 Continuity and Differentiability?
Start preparing by understanding and practicing the concept of limits. Try to be clear on the definition of continuity and differentiability and continuity and differentiability formulas. Practice many problems from each topic for better understanding. Practice continuity and differentiability class 12 previous year questions.
If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.
Important Books for Class 12 Continuity and Differentiability
Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.
Then you can refer to the book Amit M Aggarwal's differential and integral calculus or Cengage Algebra Textbook by G. Tewani but make sure you follow any one of these not all. Continuity and Differentiability are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.
A function is continuous at if the function is defined at and if the value of the function at equals the limit of the function at and a function is differentiable at a point ' ' in its domain if limit of the function exists at .
2.What are the 2 conditions of differentiability?
For a function to be differentiable, should be continuous at and the limit of the function should exist at . Fog(x) is continuous when f(x) and g(x) are both continuous.
3.What is the formula of derivative?
The formula of derivative is .
4.What are the 7 rules of differentiation?
The rules of differentiation are sum rule, difference rule, constant multiple rule, product rule, quotient rule, chain rule and power rule.
5.How to find whether the given graph is continuous?
The graph of a continuous function does not have gaps inbetween. To find whether the given graph is continuous, check if there is any gaps inbetween the graphs.