Continuity & Differentiability – Differential Calculus for GATE, ESE, AE/JE | Priyanka Ma’am

Continuity and Differentiability are core concepts of Differential Calculus and are frequently asked in GATE, ESE, AE/JE exams. These concepts help in understanding the behavior of functions and form the base for advanced topics.

Continuity

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

lim (x → a⁻) f(x) = lim (x → a⁺) f(x) = f(a)

This means:

• Left Hand Limit (LHL) = Right Hand Limit (RHL)
• Value of function exists at that point

Types of Discontinuity

• Removable Discontinuity – hole in graph
• Jump Discontinuity – LHL ≠ RHL
• Infinite Discontinuity – function tends to ∞
• Oscillatory Discontinuity – function oscillates

Differentiability

A function is differentiable at x = a if:

lim (h → 0) [f(a + h) − f(a)] / h exists

This is the derivative of the function at that point.

Relationship Between Continuity & Differentiability

• If a function is differentiable → it is continuous
• If a function is continuous → it may NOT be differentiable

Differentiability ⇒ Continuity (but not vice versa)

Important Results

• |x| is continuous everywhere but NOT differentiable at x = 0
• Polynomial functions are continuous & differentiable everywhere
• Rational functions are continuous where denominator ≠ 0

Solved Examples

Example 1

Check continuity of:

f(x) = x²

Polynomial function → continuous everywhere

Answer: Continuous

Example 2

Check differentiability of:

f(x) = |x| at x = 0

LHL derivative = -1
RHL derivative = +1

LHL ≠ RHL

Answer: Not differentiable at x = 0

Example 3

Find k for continuity:

f(x) =

k, x = 1
x², x ≠ 1

lim (x → 1) x² = 1

k = 1

Example 4

If function is differentiable at a point, then it must be:

Continuous

Shortcut Tricks

• Check LHL & RHL for continuity
• Check derivative from both sides
• |x| is most common non-differentiable example
• Use standard results directly

Questions

Q1. Differentiability implies?

Answer: Continuity

Q2. Is every continuous function differentiable?

Answer: No

Exam Tips

• Always check continuity before differentiability
• Focus on piecewise functions
• Memorize key examples like |x|
• Practice PYQs

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