[Maths Class Notes] on Continuity and Differentiability Pdf for Exam

Differentiability and Continuity is one of the most important topics and it helps students to understand various concepts like continuity at a certain point, derivative of functions, and continuity on a given interval.

 

What is Continuity?

Continuity of a function states the characteristics of the function and its functional value. A function is said to be continuous if the curve has no missing points or breaking points in a given interval or domain, that is the curve is continuous at every point in its domain.

A function f(x) is known as a continuous function at a point x = a, in its domain if the following listed three conditions are satisfied- 

     1.  f (a) exists which means that the value of f (a) is finite.

     2.  Lim x→an f (x) exists, that is the right-hand limit = left-hand limit, and both R.H.S and L.H.S are finite.

3. Lim x→an f (x)  =  f (a)

A function f(x) is said to be continuous in the given interval I that is equal to [x1, x2] only if the three conditions listed above are satisfied for every point in the given interval I.

Formal Definition of Continuity

A function is said to be continuous in the closed interval (a,b) if:

1. f is continuous in (a, b)

2. limx->a+  f(x) = f (a)

3. limx->b-  f(x) = f (b)

A function is said to be continuous in the open interval (a, b) if, f (x) is going to be continuous within the unbounded interval (a, b) if at any point within the given interval the function is continuous.

Geometrical Interpretation of Continuity

Function f is going to be continuous at x = c if there’s no break within the graph of the function at the purpose ( c , f(c) ).

In an interval, a function is claimed to be continuous if there’s no break within the graph of the function within the entire interval.

Then When can a Function be Discontinuous?

A function f is discontinuous at x=a if any of the following is true:

1. limx->a+  f(x)  and  limx->a-  f(x) exist but are not equal.

2. limx->a+  f(x) and  limx->a-  f(x) exist are both equal but not equal to f(a).

3. f (a) is not defined.

In simpler words, if the function is undefined or does not exist, then we say that the function is discontinuous.

Therefore we arrive at a modified definition of continuity: A function f is continuous at x=a if: 

 limx->a+  f(x)  =  limx->a-  f(x) = f(a)

Definition of Real Functions

A graph in the Cartesian plane can represent a real function, that is, a function from real numbers to real numbers; such a function is continuous if the graph is a single unbroken curve whose domain is the complete real line. Below is a more Mathematically rigorous definition. 

Limits are commonly used to define the continuity of real functions. If the limit of f(x), as x approaches c, is equal to f, a function f with variable x is continuous at the real number f(c).

Note:

Let f and g be two real functions and let c be a point in the common domain of f and g. If the functions f and g are both continuous at x=c then:

1. f+g is continuous at x=c.

2. f-g is continuous at x=c.

3. f*g is continuous at x=c.

4. f/g is continuous at x=c given that g(c) is not zero.

For better understanding let’s go through an Example!

What is Differentiability?

Function f(x) is said to be differentiable at the point x = a and if the derivative of the function f ‘(a) exists at every point in its given domain.

 

Differentiability Formula

The differentiability formula is defined by –

f’(a) = [frac{f(a+h)-f(a)}{h}]

If a function is continuous at a particular point then a function is said to be differentiable at any point x = a in its domain. The vice versa of this is not always true.

Here are the derivatives of the basic trigonometric functions (differentiability formulas)- 

Differentiability and Continuity Problems and Solutions-

Here are a few Differentiability and Continuity Problems and solutions!

Question 1: List down the continuity and differentiability formulas.

Solution:  The continuity and differentiability formulas are  as follows-

The differentiability problems can be solved using the formula-

f’(a) = [frac{f(a+h)-f(a)}{h}]

For a function f to be continuous it should satisfy the three conditions given below-

1.  f (a) exists which means that the value of f (a) is finite.

2.  Lim x→a f (x) exists, that is the right-hand limit = left-hand limit, and both R.H.S    and L.H.S are finite.

3. Lim x→a f (x)  =  f (a)

Question 2: Explain the Continuity of the given Function f(x). 

Where , f(x ) = sin x . cos x

Solution: We know that cos x and sin x both are continuous functions.  We also know that the product of any two continuous functions is also a continuous function.

Therefore, we can say that the function f(x) = sin x . cos x is also a continuous function.

Conclusion

One of the most significant topics is differentiability and continuity. It helps students grasp concepts such as continuity at a given point, derivatives of functions, and continuity on a certain interval.