Let f : R → R be a continuous function that is convex on (-infinity, 0] and [0, ∞) and has a local maximum at the point 0. Prove that the function f is not differentiable at the point 0.

Answer 1

Since y = f(x) is convex in the interval (-∞,0],

the limit of (y₂ - y₁) / (x₂ - x₁) > 0 for x₁ < x₂ < 0 as x₁ → 0,

i.e., f'(x) < 0 in the limit as x → 0 from below.

Similarly, since y = f(x) is convex in the interval [0,∞),

the limit of (y₂ - y₁) / (x₂ - x₁) < 0 for x₁ > x₂ > 0 as x₁ → 0.

i.e., f'(x) > 0 in the limit as x → 0 from above.

This means f'(0) is undefined, since the limit of (f(0+h)-f(0))/h is undefined as h → 0.