Differentiability and Continuity

Tags: #◇Computer Science #◇CSMATH1 #◇Problem Solving

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◇Overview

Differentiability

A function $f(x)$ is differentiable at a point $x=c$ if its derivative $f'(c)$ exists. This means the limit:
$\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$
must exist. Graphically, this means the function has a well-defined, non-vertical tangent line at the point. A function is differentiable on an interval if it is differentiable at every point in that interval.

Differentiability Implies Continuity

One of the most important theorems in calculus states that if a function is differentiable at a point, then it must be continuous at that point. The existence of a derivative requires the function to be continuous.
However, the converse is not true: Continuity does not imply differentiability. A function can be continuous at a point but fail to be differentiable there.

When does a function fail to be differentiable?

A function $f(x)$ is not differentiable at a point if its graph has any of the following features:

1. A Discontinuity: Jump, removable, or infinite discontinuity. If it's not continuous, it cannot be differentiable.
2. A Sharp Corner or Cusp: The slope of the curve approaches different values from the left and the right (e.g., the absolute value function $f(x) = |x|$ at $x=0$).
3. A Vertical Tangent: The curve goes straight up or down, making the slope undefined (approaching infinity).

◇Important Formulas

Differentiability Condition: $f'(c)$ exists if and only if $\lim_{h \to 0} \frac{f(c+h) - f(c)}{h} = L$ for some real number $L$.

◇Worked Examples

Example 1: Continuous but not Differentiable
Consider the absolute value function $f(x) = |x|$.
Continuity: The function is continuous everywhere because its graph can be drawn without lifting the pencil.
Differentiability: At $x = 0$, the function comes to a sharp corner.
The left-hand derivative (slope for $x < 0$) is $-1$.
The right-hand derivative (slope for $x > 0$) is $1$.
Because the limits from the left and right are different, the derivative at $x = 0$ does not exist. Thus, $f(x) = |x|$ is continuous but not differentiable at $x=0$.

◇References & Navigation

• Previous Topic: ◇The Tangent Line and the Derivative
• Next Topic: ◇Differentiation Rules
• Related Notes: ◇Continuity, ◇The Tangent Line and the Derivative