Differentiability

Understanding Differentiability

A function $f$ is differentiable at $x=a$ if the derivative $f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}$ exists as a single, finite number. Graphically, this means the curve has one well-defined, non-vertical tangent line at that point.

Three Ways Differentiability Fails

Corners: The graph approaches a point with different slopes from left and right. The classic example is $f(x) = |x|$ at $x=0$. From the right, $f'(x) = 1$; from the left, $f'(x) = -1$. Since $\lim_{h\to 0}\frac{|h|}{h}$ yields $1$ for $h>0$ and $-1$ for $h<0$, the two-sided limit does not exist.

Cusps: One-sided slopes blow up to opposite infinities from both sides, pinching the graph to a sharp point. For $f(x) = x^{2/3}$ at $x=0$, the slope approaches $+\infty$ from the right and $-\infty$ from the left, so $f'(0)$ does not exist.

Vertical Tangents: The graph is smooth but the tangent line becomes vertical—the slope approaches the same infinity from both sides. For $f(x) = \sqrt[3]{x}$ at $x=0$, we have $f'(x) = \frac{1}{3}x^{-2/3}$, so $f'(0) = +\infty$ from both sides. The limit is infinite, not finite, so the function is not differentiable.

Critical Insight

Differentiability implies continuity, but continuity does not imply differentiability. A function can be continuous everywhere yet fail to be differentiable at points with sharp features. Always check the limit definition when a graph looks sharp or vertical.