Is this a way to show a series does not converge uniformly?

Question

So (correct me if I'm wrong please), this following is a fact:
For a sequence of $\textbf{continuous}$ functions $\{f_n(x)\}_{n=1}^{\infty}$, the series $\sum_{n=1}^{\infty} f_n(x)$ converges uniformly to $f \implies$ $f$ is continuous.

So the negation of that (please correct me if I'm wrong):
$f$ is not continuous $\implies$ the sequence $\{f_n(x)\}_{n=1}^{\infty}$ are $\textbf{discontinuous}$ functions $\textbf{or}$ the series $\sum_{n=1}^{\infty} f_n(x)$ does not converge uniformly.

I wonder if my logical negation in the second step is 100% correect (and if the original statement is true)

Answer 1

The conclusion should be either the series is not uniformly convergent or that at least one of the $f_n$ is discontinuous.