Show $g(x)=\frac{f(x+x_0)-f(x_0)}{x}$ is Bounded in some nbd of $x_0$

Question

let $f:[a,b]$ and $f$ is continuous at $x_0 \in[a,b]$ and there is some $\delta>0$ with the property that $f$ has a bounded derivative at all points in $(x_0-\delta,x_0+\delta)/\{x_0\}$ i.e $sup_{y\in (x_0-\delta,x_0+\delta)/\{x_0\}} |f'(y)|< \infty$.

let $g(x)=\frac{f(x+x_0)-f(x_0)}{x}$

then we have to show that $g(x)$ is bounded in some neighbourhood of $x_0$

how to approach this problem.$f$ is bounded in some nbd of $x_0$ because of continuity. i was thinking of MVT is some nbd of $x_0$, but could not relate it to problem.

any hint please

Answer 1

Let $C:=sup_{y\in (x_0-\delta,x_0+\delta)/\{x_0\}} |f'(y)|$ . If $x$ has the property that $x+x_0 \in (x_0-\delta,x_0+\delta)$, then, by the Mean Value Theorem, there is $s$ between $x_0$ and $x+x_0$ such that

$g(x)=f'(s)$.

Hence $|g(x)| \le C$.