Differentiability Implies Continuity

Real Analysis · Axiom Academy

EXAMPLE Differentiability Implies Continuity A rigorous proof using limit properties showing why a function must be continuous at any point where it is differentiable. Prove the following theorem: if f is differentiable at a , then f is continuous at a . We will build the proof one step at a time, choosing the right move at each stage. Nice work — you built a rigorous proof of a fundamental theorem in real analysis. Here's what made it work: Differentiability is stronger than continuity: if a function has a derivative at a point, it must be continuous there. The key algebraic trick: writing f(x) - f(a) as exposes the difference quotient so we can use differentiability. Limit laws are essential: the product rule for limits lets us separate the two factors, since both limits exist. The converse is false: a function can be continuous at a point without being differentiable there — for example, |x| is continuous at 0 but its one-sided slopes are -1 and +1 , so the derivative does not exist. Why derivatives need continuity: the limit of the difference quotient cannot exist if the function has a jump, so differentiability forces continuity. The hierarchy is one-directional: differentiability continuity, but continuity differentiability. Keeping that arrow straight is essential throughout real analysis.

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