Chapter 5 Continuity and Differentiability

Notes

Continuity and Differentiability

1. Continuity

Let f(x) be a real-valued function defined on its domain. The function f(x) is said to be continuous at a given point x = a if:

lim_x→a f(x) = f(a)

For a function to be continuous at x = a:

• Right Hand Limit (RHL) must exist.
• Left Hand Limit (LHL) must exist.
• Functional value f(a) must exist.
• RHL = LHL = f(a).

Condition of Continuity:

RHL = LHL = f(a)

Right Hand Limit (RHL)

To find RHL at x = a, put:

x = a + h

When x → a⁺, h → 0.

Therefore,

RHL = lim_h→0 f(a+h)

Left Hand Limit (LHL)

To find LHL at x = a, put:

x = a - h

When x → a^-, h → 0.

Therefore,

LHL = lim_h→0 f(a-h)

Discontinuous Function

If the condition

RHL = LHL = f(a)

is not satisfied, then the function is said to be discontinuous at x = a.

2. Continuity of Polynomial Functions

Every polynomial function is continuous for all real values of x.

For example, for f(x) = x² at x = 2:

f(2) = 4

RHL = lim_h→0 (2+h)² = 4

LHL = lim_h→0 (2-h)² = 4

Therefore,

RHL = LHL = f(2) = 4

Hence, f(x) = x² is continuous at x = 2.

Similarly, for f(x) = x³ at x = 2:

f(2) = 8

RHL = 8 and LHL = 8.

Therefore, f(x) = x³ is continuous at x = 2.

3. Greatest Integer Function

The greatest integer function [x] is discontinuous at integral points.

Let x = a, where a is an integer.

Functional value:

[a] = a

Right Hand Limit:

lim_h→0 [a+h] = a

Left Hand Limit:

lim_h→0 [a-h] = a-1

Since RHL ≠ LHL, the greatest integer function is discontinuous at integral points.

4. Modulus Function

The modulus function |x| is continuous for every real value of x.

5. Algebra of Continuous Functions

Let f(x) and g(x) be two continuous functions. Then, on their common domain:

• f + g is continuous.
• f - g is continuous.
• fg is continuous.
• f/g is continuous wherever g ≠ 0.

6. Continuity of Trigonometric Functions

Continuity of sin x

Let f(x) = sin x.

At x = a:

f(a) = sin a

RHL = lim_h→0 sin(a+h)

Using sin(a+h) = sin a cos h + cos a sin h: