For a continouous function limsup and liminf are equal

Question

I need to prove continuity of a function by showing that for a bounded function $f : [a,b] \rightarrow \mathbb{R}$

$lim_{\delta \rightarrow 0}Sup_{|y-x| \leq \delta} f(y) =H(x) $ and $lim_{\delta \rightarrow 0}Inf_{|y-x| \leq \delta} f(y) =h(x) $

then prove that $H(x) = h(x) $ iff f is continuous at $X$

How should i proceed ?

Answer 1

If $f$ is continuous at a point $x \in [a,b]$, then $\displaystyle{\lim_{y \to x} f(y)=f(x)}.$ In particular, since the limit exist at $x$, it means that the lim sup and the lim inf agree.

On the other hand, if $H(x)=h(x)$, then $\displaystyle{\lim_{y \to x}} f(y) =f(x),$ hence your function is continuous at $x.$