The Weierstrass function W(x) = sum(n=0 to inf) $a^n$ cos($b^n$ pi x) (with 0 < a < 1, ab > 1+3pi/2) is continuous everywhere but differentiable nowhere.

This shows that continuity is a strictly weaker condition than differentiability — there exist functions that are continuous at every point but differentiable at no point.

While not directly tested in JEE, this concept appears in true/false type questions: "Every continuous function is differentiable at at least one point" — FALSE.