Suppose for exemple that $f(x)$ is an increasing function (take $-f$ if $f$ is decreasing). First of all : $f(x_1)(x_2-x_1) \leq \int_{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) \leq f(x_2)(x_2-x_1)$ , so there is a such $c\in ]x_1,x_2[$ Let $u,r,t$ be such that $u\in ]a,b[ , 0 < r < 1, 0 < t < \frac{\min (u-a,b-u)}2 $ We can write $f(c_{(rt)})*(r+1)t = \int _{u-t}^{u+rt}f(x)dx = $ $\int_{u-t}^{u}f(x)dx + \int_{u}^{u+rt}f(x)dx = f(p_{(t)})*t + f(q_{(rt)})*t$ , so $f(c_{(rt)})=\frac {f(p_{(t)})+rf(q_{(rt)})}{1+r}$, where $c_{(t)}\in ]u-t,u+t[ \ ,\ u-t < p_{(t)} < u < q_{(rt)} < u+rt$ If $t \longrightarrow 0$ then $p_{(t)}\longrightarrow u \ ,\ q_{(rt)}\longrightarrow u$ , so $f(p_{(t)}) \longrightarrow f_{u-} \ , \ f(q_{(rt)}) \longrightarrow f_{u+}\ $ (left and right limits of $f$) , so $f(c_{(rt)})\longrightarrow \frac {f_{u-}+rf_{u+}}{1+r}\ ,\ f_{u-}\leq f(u) \leq f_{u+}$ , . But $\ c_{(rt)}\in ]u-t,u[\ \cup \ {u\}\ \cup \ ]u,u+rt[}$ $\Longrightarrow f(c_{(rt)})\in\ ]f(u-t),f(u)[\ \cup\ \{f(u)\}\ \cup \ ]f(u),f(u+rt)[$ So for the limit of $f(c_{(rt)})$ we have :$ \frac {f_{u-}+rf_{u+}}{1+r} \in \{f_{u-},f(u), f_{u+}\} , \forall 0<r<1$ It's easy to see that : $\frac {f_{u-}+rf_{u+}}{1+r}=f_{u-} \Longrightarrow f_{u-}=f_{u+} \ ;\ \frac {f_{u-}+f_{u+}}{1+r}=f_{u+} \Longrightarrow f_{u-}=f_{u+}$ Suppose that $f_{u-} \neq f_{u+}$ , so $\frac {f_{u-}+rf_{u+}}{1+r}= f(u), \forall 0<r<1\ $, so $f_{u-} = f_{u+}$ $f_{u-} = f_{u+} \Longrightarrow f_{u-} = f_{u+} =f(u) \Longrightarrow \ continuity $

If $f$ is not monotonic, the result does'nt hold : take for exemple $f(0)=0 \ , \ f(x)=1 \forall x\neq 0$

nice solution ,thank a lot