Prove that if a function is increasing on [a,b] and satisfied the intermediate value property, then that function is continuous on [a,b].

Let $f$ be increasing on $[a,b]$, i.e. for all $x<y$ in $[a,b]$, $f(x)\leq f(y)$. Also assume $f$ satisfies the intermediate value property. Show that $f$ is continuous on $[a,b]$.

My attempt: Let $f$ be increasing on $[a,b]$ and satisfy IVP, i.e. $\forall x<y$ on $[a,b]$ and $\forall L$ between $f(x)$ and $f(y)$, $\exists c\in(x,y)$ with $f(c)=L$.

Let $\epsilon>0$. Let $c\in(a,b)$. To show $f$ continuous on $[a,b]$, we need to choose a $\delta>0$ such that whenever $|x-c|<\delta$, $|f(x)-f(c)|<\epsilon$.

Since $f$ is increasing and $a<c$, then $f(a)\leq f(c)$. I want to show that $f(c)-\epsilon\leq f(x)\leq f(c)+\epsilon$ but don't know how.