calculus - prove that a function whose derivative is bounded also bounded

I got this problem:

Let $f$ be a differentiable function on an open interval $(a,b)$ such that $f'$ (the derivative of $f$) is bounded on $(a,b)$ (meaning there exist $0

I tried to prove it but wasn't able to proceed.

Thanks.

Answer

Fix a point $x_0\in (a,b).$ Assume $x\in(x_0,b).$ By using the Lagrange's theorem there exists $c\in(x_0,x)$ such that $f(x)-f(x_0)=f'(c)(x-x_0).$ Thus

$$|f(x)|=|f(x_0)+f'(c)(x-x_0)|\leq |f(x_0)|+|f'(c)||(x-x_0)|\leq |f(x_0)|+M(b-a).$$ Proceeding in the same way you get the bound for $x\in(a,x_0).$ Thus we have shown that the function is bounded.