Continuity and differentiability are fundamental concepts in calculus, forming the backbone of mathematical analysis. Understanding these concepts is essential for students as they provide the foundation for more advanced topics in calculus such as integration and differential equations.

This article provides an overview of continuity, differentiability, and important formulas and concepts. Additionally, it includes practice questions with solutions.

What is Continuity and Differentiability?

Continuity

The continuity of a function means that its graph is unbroken; there are no jumps, gaps, or holes in it. A function f(x) is continuous at a point x = a if the following three conditions are met:

• The function is defined at x = a: f(a) exists.
• The limit of the function as x approaches a exists: lim⁡_x→af(x)exists.
• The value of the function at x = a is equal to the limit as x approaches a: lim⁡_x→af(x) = f(a).

For a comprehensive understanding, you can refer to Continuity.

Differentiability

Differentiability refers to the existence of the derivative of a function at a point. A function f(x) is differentiable at x=a if the derivative f′(a) exists. The derivative of f(x) at x=a is defined as :

[Tex]f'(a) = \lim_{h \to a} \frac{f(a + h) – f(a)}{h} [/Tex]

If a function is differentiable at a point, it is also continuous. However, the converse is not always true—a function can be constant at a point but not differentiable there.

For more details, check out Differentiability.

Related Formulas / Concepts

Continuity Formula

A function f(x) is continuous at x = a if:

Iim_x→a f(x) = f(a)

Intermediate Value Theorem: If f(x) is continuous on [a,b] and k is any number between f (a) and f(b) then there is at least one number c in [ a,b] such that f(c) = k.

Differentiability Formulas

A function f(x) is differentiable at x = a if:

[Tex]f'(a) = \lim_{h \to 0} \frac{f(a+h) – f(a)}{h} [/Tex]

Chain Rule: If y = f(u) and u = g(x) , then :

[Tex]\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} [/Tex]

Product Rule: If u(x) and v(x) are differentiable functions, then:

(uv)′ = u′v + uv′

Quotient Rule:If u(x) and v (x) are differentiable functions, then:

[Tex]\left( \frac{u}{v} \right)’ = \frac{u’v – uv’}{v^2} [/Tex]

Practice Questions on Continuity and Differentiability: Solved

1. Determine if the function f(x) = x² – 4 / x-2 is continuous at x = 2.

Simlify the function

f(x) = (x-2) (x+2) / x-2 = x+2 (for x ≠ 2)

lim_x→2 f(x) = lim_x→2​ (x+2 ) = 4

since f(2) is undefined f(x) is not continuous at x=2 .

2. Show that the function f(x) = x² is continuous at x = 1.

lim​_x→1f(x) = lim_x→1​ x² = 1

f(1) = 1

Since lim⁡_x→1f(x) = f(1) , the function is continuous at x=1

3. Show that the function f(x) = x² is continuous at x=1.

Iim _x→1 f(x) = lim _x→1 x² = 1

f(1) = 1

Since lim _x→1 f(x) = f(1) , the function is continuous at x=1

4. Evaluate lim_x→3 x² – 9 / x-3.

x²-9 / x-3 = (x-3) (x+3) / (x-3) = x+3 (for x is not equal to 3 )

lim_x→3 (x+3) = 3+3 = 6

5. Show that f(x) = 3x+2 is continuous at x = -1 .

lim_x→1 f(x) = lim_x→1 (3x+2 ) = 3(-1) + 2 = -3 + 2 = -1

f(-1) = 3(-1) + 2 = -3+2= -1

since lim_x→1 f(x) = f(-1), the function f(x) = 3x + 2 is continuous at x = -1

6. Show that f(x) = 1/x is continuous at x≠ 0.

For x ≠ 0

lim_x→a f(x) = lim_x→a 1/x = 1/a

f(a) = 1/a

since lim_x→a f(x) = f(a) for all a ≠ 0 , f(x) is continuous for x ≠ 0.

7. Check if f(x) = x ³ is continuous at x = 0

lim _x→0f(x) = lim _x→0 x³ = 0

f(0) = 0

since lim _x→0f(x) = f(0) , the function is continuous at x = 0.

8. Determine if the function f(x) = e^x-1 / x is continuous at x = 0.

Solving using L Hospital Rule:

Iim_x→0 e^x -1 / x = lim _x→0 (e^x/1) = 1

f(0) = 1

since Iim_x→0f (x) = f (0) the function is continuous at x=0

9. Show that f(x) = cos (x) is continuous for all x.

For any a,

lim_x→a cos (x) = cos (a)

f(a) = cos (a)

since lim_x→af (x) = f (a), the function is continuous for all x .

10. Show that f(x) = x²-4 / x-2 is continuous for x ≠ 2

f(x) = (x- 2)(x+2) / x-2 = x+2 (for x ≠ 2 )

For x ≠ 2

lim_x→af(x) = a+2

f(a) = a + 2

since lim_x→af (x) = f(a) for all a≠ 2, f (x) is continuous for x ≠ 2.

Practice Questions on Continuity and Differentiability: Unsolved

Below are the practice questions on Continuity and Differentiability are as following :

Related Articles:

• Continuity and Differentiability
• Continuity of Function
• Differentiability of a Function

Continuity and Differentiability – FAQs

What is the difference between continuity and differentiability?

Continuity refers to the unbroken nature of a function’s graph, while differentiability refers to the existence of a function’s derivative at a point. Differentiability implies continuity but continuity does not imply differentiability.

Can a function be continuous but not differentiable?

Yes, a function can be continuous but not differentiable. For example, the absolute value function f(x) = ∣x∣ is continuous at x = 0 but not differentiable there.

What is the mean value theorem for continuity and differentiability?

For any continuous function, f(x) is continuous on [a, b] and differentiable on (a, b), According to the Mean Value Theorem, there exists a ‘c’ in the interval (a, b) such that f'(c) = [ f(b) – f(a) ] / (b – a).

Why is continuity important in calculus?

Continuity is important because many theorems in calculus, such as the Intermediate Value Theorem and the Fundamental Theorem of Calculus, require functions to be continuous.

What is the relationship between continuity and differentiabilities of a function?

A continuous function can be non-differentiable. Any differentiable function is always continuous. However, a continuous function does not have to be differentiable. Any function on a graph where a sharp turn, bend, or cusp occurs can be continuous but fails to be differentiable at those points.