Differentiability

Core Definition

A function $f(x)$ is differentiable at a point $x = a$ if the derivative $f'(a)$ exists:

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \quad \text{exists and is finite.}

A function is differentiable on an open interval

(a, b)

if it is differentiable at every point in the interval.

Concept Explanation

The Differentiability Implies Continuity Theorem

A fundamental theorem of calculus establishes:

\text{If } f(x) \text{ is differentiable at } x = a, \text{ then } f(x) \text{ is continuous at } x = a.

The Converse Is False!

Continuity does NOT guarantee differentiability. A graph can be completely unbroken and continuous, yet fail to possess a derivative at a specific point.

The Four Visual Signatures of Non-Differentiability

A function fails to be differentiable at $x = a$ if any of the following four conditions occur:

Corner / Sharp Turn: The function is continuous, but the one-sided slopes from the left and right do not match: $\lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h} \neq \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$ Classic example: $f(x) = |x|$ at $x = 0$ .
Cusp: An extreme sharp point where the curve becomes infinitely steep from both sides with opposite infinite signs ( $+\infty$ from one side, $-\infty$ from the other). Classic example: $f(x) = x^{2/3}$ at $x = 0$ .
Vertical Tangent: The curve becomes infinitely steep for a single instant, causing the slope limit to become $\pm\infty$ . Since $\infty$ is not a real number, the derivative does not exist. Classic example: $f(x) = x^{1/3}$ at $x = 0$ .
Discontinuity: If $f(x)$ has a removable hole, jump, or vertical asymptote at $x = a$ , no unique tangent line can be constructed. Non-continuity automatically implies non-differentiability.

Engineering Applications

Civil & Transportation Engineering: Highway exit ramps, railway transitions, and roller coaster tracks must be designed with $C^1$ -smooth continuity (continuous first derivative). A sharp corner (non-differentiable transition) would cause an instantaneous impulse spike in lateral centrifugal acceleration, leading to vehicle derailment or passenger injury.
Mechanical Stress Concentrations: Internal geometric corners in load-bearing structural brackets produce non-differentiable stress profiles, concentrating forces at the vertex. Mechanical engineers introduce smooth transition curves (fillets) to ensure differentiability and prevent fatigue fractures.

Example Problems

Example 1: Piecewise Boundary Differentiability Check

Determine if $f(x)$ is differentiable at $x = 2$ :

f(x) = \begin{cases} x^2 & x \le 2 \\ 4x - 4 & x > 2 \end{cases}

Solution

Step 1: Check Continuity at $x = 2$ :
- Left limit: $\lim_{x \to 2^-} x^2 = 4$
- Right limit: $\lim_{x \to 2^+} (4x - 4) = 4(2) - 4 = 4$
- Function value: $f(2) = 4$ Since $\lim_{x \to 2} f(x) = f(2) = 4$ , $f(x)$ is continuous at $x = 2$ .
Step 2: Check One-Sided Derivatives at $x = 2$ :
- Left-hand derivative: $\frac{d}{dx}[x^2]\Big|_{x=2} = 2x\Big|_{x=2} = 4$
- Right-hand derivative: $\frac{d}{dx}[4x - 4]\Big|_{x=2} = 4$
Conclusion: Since $f(x)$ is continuous at $x = 2$ and the left-hand derivative equals the right-hand derivative ( $4 = 4$ ), $f(x)$ is differentiable at $x = 2$ .

Connections & References

Parent Concepts: Continuity, Limits
Related Concepts: Derivatives, Discontinuities, Piecewise Functions
Course Sessions: Calc-1 Session 06