If $f_n \to f$ is uniformly, does that mean $f$ is continuous? As I was studying uniform continuity, a question hit me:

If $f_n \to f$ is uniformly, does that mean f is continuous?

I think the answer is no, but i'm not sure. Can someone help me?

As Alex said, if the $f_n$ are discontinuous this is false. But, the uniform limit of  continuous functions is continuous.
Let $\{f_n\}_{n \in \mathbb{N}}$ be a sequence of functions converging uniformly to $f$. Fix $\epsilon > 0$. Then, by uniform convergence there exists $N > 0$ such that $$|f(x)-f_n(x)| < \epsilon/3$$ for $n > N.$ By continuity of $f,$ $$|f_N(x)-f_N(y)| < \epsilon/3.$$ By the triangle inequality $$|f(x)-f(y)| < |f(x)-f_N(x)| + |f_N(x)-f_N(y)| + |f(y)-f_N(y)| < \epsilon.$$
