Non-Differentiable Functions

Real Analysis · Axiom Academy

LESSON Non-Differentiable Functions Exploring where derivatives fail to exist: corners, cusps, vertical tangents, and jumps reveal the subtle requirements for differentiability. 1. What Differentiability Demands A function f is differentiable at x = a when the difference-quotient limit below exists — and that limit, when it exists, is the derivative f'(a) . Geometrically it is the slope of a single, non-vertical tangent line at the point. Our first failure is the cleanest. The function f(x) = |x| is continuous everywhere, yet it makes a sharp turn at x = 0 . Watch a secant approach from the left and another from the right: each one settles on a definite slope, but the two slopes never agree. The function f(x) = x^ 2/3 has a cusp at x = 0 . Unlike a corner, both branches reach the same point — but they arrive with slopes that grow without bound, steepening toward the vertical from opposite sides. As , the factor depending on the side. The function f(x) = x^ 1/3 — the cube root — passes smoothly through x = 0 with no corner and no cusp. Yet it still fails to be differentiable, because its tangent line there is vertical . As , the factor from both sides. 4. Jumps: Differentiable Implies Continuous There is one failure that needs no secant chase at all. If a function is differentiable at a point, it must be continuous there — so the contrapositive says any discontinuity rules out differentiability immediately. A jump forces the difference quotient to blow up.

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