Continuity and Differentiability

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Introduction:
Students in Class 12 Maths are learning about Continuity and Differentiability for the first time. The complexity of this Chapter varies from moderate to challenging since difficulties may be readily answered once students have completed the formulas. However, the hardest part comes in their tests, because this Chapter has a large number of questions, the bulk of which are in the form of application kinds.If the function is defined at x = c .Because the function's value at x = c equals the function's limit at x = c, the function is continuous at x = c. If function f is not continuous at c, then c has termed a point of discontinuity of function f. The feature of continuity may be seen in many aspects of nature. There is a continual flow of water in the rivers. Because human existence is a continuous passage of time, you will constantly get older. The list goes on. Similarly, we have the idea of function continuity in mathematics. Simply said, a function's curve is deemed continuous if you can write it on a graph without raising your pen even once (provided that you can draw well). Actually, it's a really straightforward and true definition. However, for the sake of advanced mathematics, we must define it more accurately.

List of topics according to NCERT and JEE Main/NEET syllabus:

Algebra of a continuous function
Derivatives of composite functions
Derivatives of implicit functions
Derivatives of inverse trigonometric functions
Exponential and Logarithmic Functions
Logarithmic Differentiation
Derivatives of Functions in Parametric Forms
Second-Order Derivative
Mean Value Theorem

Important concepts and Laws:

The definition of the term "continuity" is the same as what we use in our everyday lives. The water flow in rivers, for example, is constant. The passage of time in human existence is constant, i.e. we continue to age and so on. Similarly, in mathematics, we have the concept of function continuity.

When we claim that a function f(x) is continuous at x = a, we imply that the graph of the function has no holes or gaps at the point (a, f(a)). In basic terms, a function is considered to be continuous if its curve can be sketched on a graph without ever raising your pen.

Let c be a point in the domain of f and f be a real function on a subset of the real numbers.

Lim x → c f(x) = f(c)

Let f and g be two continuous real functions at a real integer c.

At x = c, f + g is continuous.
At x = c, f - g is continuous.
At x = c, f * g is continuous.
At x = c, where g(c) 0, f/g is continuous.

A one-real-variable differentiable function is one whose derivative exists at each point in its domain. As a result, each point in the domain of a differentiable function's graph must have a tangent line. There should be no fractures, bends, or cusps, and it should be rather smooth.

The theorem of Mean Value

In Calculus, there are two fundamental results. Both will be discussed, as well as the geometric interpretations of these theorems.

NCERT Notes Subject wise link:

Theorem of Rolle:

Let f be a real-valued function that is defined on the closed interval [a, b] and has the property that

On the closed interval [a, b], it is continuous.

On the open interval, it is differentiable (a, b)

f(a) = f(b) = f(c) = f(d) = f(e) = f( (b)

Then a real integer c (a, b) exists such that f′(c) = 0.

Inverse Trigonometric Functions: The class of inverse functions is quite broad, and it is responsible for accomplishing the inverse of what a function performs, as the name implies. The multiplication function, for example, is the inverse of the division function. Because of their broad relevance, it's critical to comprehend their continuous and differentiable character across domains.

NCERT Solutions Subject wise link:

NCERT Exemplar Solutions Subject wise link:

Frequently Asked Question (FAQs) - Continuity and Differentiability

Question: What is the relationship between differentiability and limit continuity?

Answer:

Differentiability is defined as when the slope of the tangent line equals the function's limit at a particular location. This implies that a function must be continuous in order to be differentiable, and its derivative must be continuous as well.

Question: How to differentiate logarithmic and exponential Functions?

Answer:

d log x /dx = 1/x

dex/dx = ex can be used to distinguish between exponential and logarithmic functions.

Question: What is the best way to locate the point of discontinuity?

Answer:

The point of discontinuity in most issues would be pointing a,b, etc., where Function is of form.

F(x) = f1(x) b>x>a

f1(x) x ≤ a so on

Question: When does the function fog continue?

Answer:

Fog(x) is continuous when f(x) and g(x) are both continuous.

Question: What is the rule of L'Hospital?

Answer:

L'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if

Lim x → c f(x) = Lim x → c g(x) = 0 or 土∞ and g’(x)≠0 for all x in I with x ≠ c, and Lim x → c f(x)g(x)exists,

Then Lim x → c f(x)g(x) = Lim x → c f'(x)g'(x)