← Calculus methods

Points where a function is not differentiable

A less common but real exam topic: a function built from two different pieces meeting at a sharp corner, where the ordinary derivative rules break down.

What makes a point non-differentiable

  • At a SHARP CORNER (like the vertex of y = |x|), the gradient approaching from the left and the gradient approaching from the right are DIFFERENT — there is no single, well-defined tangent line at that exact point.
  • The gradient FUNCTION itself has a discontinuity (a jump, gap or break) at that x-value, even though the original function itself may be perfectly continuous there.
  • A graphing convention: mark the gradient function with an OPEN CIRCLE at the x-value where it is undefined, showing the value is approached from both sides but never actually reached at a single point.
Worked example Why y = |x| has no single gradient at x = 0 Explain why the function y = |x| is not differentiable at x = 0. For x < 0, y = −x, so the gradient is −1 everywhere on that side. For x > 0, y = x, so the gradient is +1 everywhere on that side. Approaching x = 0 from the left gives a gradient of −1; approaching from the right gives a gradient of +1 — these do not match. Since there is no single value the gradient approaches from both directions, the function has no well-defined tangent line, and no derivative, exactly at x = 0. ✓ The gradient jumps instantaneously from −1 to +1 at x = 0, so no single tangent line — and therefore no derivative — exists at that exact point, even though y = |x| itself is a perfectly continuous, unbroken curve there. ✕

A continuous function is not automatically a differentiable one

y = |x| has no gaps or jumps in its own graph — it is perfectly continuous everywhere, including at x = 0. But continuity and differentiability are DIFFERENT properties: a sharp corner can exist on an otherwise unbroken curve, and at that corner the function fails to be differentiable even though it never stopped being continuous. Do not treat "the graph has no gaps" as proof that a derivative exists everywhere.

Practise just this