How to prove that $f$ is continuous?

Question

$f,g:R->R$ with
$g(x)=f(f(x)) + e^x$ , $xεR$
supposed $f$ , $g$ are strictly increasing and for $x_0εR$ , $lim_(xrarrx_0)f(x)=l$

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Answer 1 · 2017-12-05T13:33:41

$lim_(xrarrx_0)f(x)=f(x_0)=L$

$lim_(xrarrx_0)f(x)=L$ $<=>$ $lim_(xrarrx_0^+)f(x)$ = $lim_(xrarrx_0^-)f(x)=L$

If $x<$ $x_0$ $=>$ $x->x_0^-$ $=>$ $f(x)$ $<$ $f(x_0)$ , because $f$ strictly increasing

so $=>$ $lim_(xrarrx_0^-)f(x)$ $<=$ $f(x_0)$ $(1)$

If $x>$ $x_0$ $=>$ $=>$ $x->x_0^+$ $=>$ $f(x)$ $>$ $f(x_0)$ , because $f$ strictly increasing

so $=>$ $lim_(xrarrx_0^+)f(x)$ $>=$ $f(x_0)$ $(2)$

$L<=f(x_0)$
&
$L>=f(x_0)$

$=>$ $f(x_0)=L$