Assumptions that imply $\sup_{c \in [a_k,b]} f(c,x) \to f(b,x)$ if $a_k \to b$

Question

Suppose $f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} $ is continuous and with compact support. What assumptions imply $$\sup_{c \in [a_k,b]} f(c,x) \to f(b,x)$$ as $a_k \to b$? How can I prove it?

Answer 1

It is not clear how $f$ is defined, but I assume it is on $\mathbb{R}^2$. Fix a real $x$. Then for each $I_k :=[a_k,b]$, there is a point $c_k \in I_k$ for which $$ f(c_k,x) = \sup_{c\in I_k} f(c,x). $$ It is clear that $c_k$ converges to $b$ since $\bigcap_k I_k = \{b\}$. By continuity of $f$ (in first argument), $$ \lim_{k\to\infty} \sup_{c \in I_k} f(c,x) = \lim_{k\to\infty} f(c_k,x) = f(b,x). $$