Uniform Convergence continuous

Question

True or False:

"If $(f_n)$, where each $f_n$ is continuous, converges to $f$ on $S$ and $f$ is not continuous on $S$, then the convergence is not uniform.

This statement seem false to me, since I know:

"If a sequence of continuous functions $\{fn\}$ converges pointwise to a function $f$, $f$ is not necessarily continuous."

Can anyone confirm this?

Answer 1

The statement is true, because it is proven in Real Analysis courses that if a sequence $(f_n)_{n\in\mathbb N}$ of continuous functions converges uniformly to a function $f$, then $f$ is continuous too. Your argument proves nothing.

Answer 2

There is a theorem that states that if a sequence of continuous functions converges uniformly, then it converges to a continuous function (I.E - the metric space of continuous functions is complete under the uniform convergence norm).

Since we have a sequence of continuous functions that converges (point-wise!) to a non-continuous function, the convergence cannot be uniform.