# Counterexample to Uniform Convergence of Convex Functions

It is stated here that if a sequence $$f_n : [a,b] \to \mathbb{R}$$ of convex continuous functions converges pointwise to a continuous function $$f$$, then the convergence is in fact uniform.

The answers given seem me to be correct proofs of the statement, however I was able to construct a counterexample pretty easily. Consider the functions

$$\begin{equation*} f_n = \begin{cases} -nx, & \text{if } 0 \leq x \leq \frac1n, \\ -2 + nx, &\text{if } \frac1n \leq x \leq \frac2n,\\ 0, &\text{otherwise}. \end{cases} \end{equation*}$$ It's clear that $$f_n$$ is convex and continuous, and moreover converges pointwise to $$f \equiv 0$$. However, the convergence is not uniform, since $$\sup |f_n| = 1$$.

I'm not sure whether there's I overlooked something when constructing this counterexample or the original statement about uniform convergence of convex functions is incorrect.

Try plotting your functions, are they actually convex? (Hint: you need to specify the domain of your $$f_n$$)