If $f$ is continous at $c$, prove $\lim_{h \to 0} (\inf \,\{f(x)\mid c \leqslant x \leqslant c+h\})=f(c)$(duplicated) Let $f$ be continous at $c$. Prove $$\lim_{h \to 0} \left(\inf \,\{f(x)\mid c \leqslant x \leqslant c+h\}\right)=f(c)$$ 
This fact is used in Spivak's book to prove 1nd Fundamental Calculus Theorem.
This question is duplicated, I have done it previously but i didn't understand yet

The inf is bounded from below for every $h > 0$ because f is continous (can you show why?). Assuming
$$
\lim_{h \rightarrow \infty} \inf \{ f(x) : c \leq x \leq c + h \} =: K \neq f(c),
$$
then there exits some sequence $(x_n)_n$ such that for all $n \in \mathbb{N} : x_n > c, \lim_{n \rightarrow \infty} x_n = c$ and $\lim_{n \rightarrow \infty} f(x_n) = K \neq f(c) = f(\lim_{n \rightarrow \infty} x_n)$ a contradiction to the continuity of $f$.
