Does a differentiable function on $[a,b]$ has bounded variation ? I recall that differentiable on $[a,b]$ means differentiable on $(a,b)$ and $\lim_{x\to a^+}f(x)$ and $\lim_{x\to b^-}f(x)$ exist.
I know that if $f'$ is integrable on $[a,b]$, then it works (since we can majorate $V_{[a,b]}(f)$ by $\int_a^b|f'|$). But if $f'$ is not integrable ? Can we find a differentiable function on $[a,b]$ that has no bounded variation ?
Best Answer
A typical example is this:
Take the function $f(x)=x^2 \sin{\frac{1}{x^2}}$ if $x \neq 0$ and $f(0)=0$ on $[0,1]$