Is the following statement true? fₙ → f uniformly ⇒ fₙ′ → f′

No. fₙ(x) = sin(nx)/n Converges uniformly to 0, derivatives cos(nx) do not converge.

Counterexample

Is this true?

$f_{n} \to f$ uniformly $\Rightarrow f_{n}' \to f'$

No — it is false.

The counterexample

$f_{n}(x) = \sin($ nx)/n

Why it fails

Converges uniformly to 0, derivatives cos(nx) do not converge.

Keep practicing

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.