condition of changeability of function

Question

Let $(f_n)$ be a sequence in $\Bbb R \to \Bbb R$ that converges to a continuous function $f(x)$. Is it true that $\lim_{x \to a} f(x) = \lim_{n \to \infty} f_n (a)$?

Answer 1

Since $f$ is continuous you have $\lim_{x \to a}f(x)=f(a)$ by definition. And also the definition of $f_n \to f$ (pointwise, not necessarily uniformly) implies that at each $x$, in particular at $a$, $f_n(a) \to f(a).$