If a function $f$ is differentiable on $[a,b]$ and its derivative $f^\prime$ is integrable, must $f$ be of bounded variation?

Question

I know there is a theorem saying if $f$ defined on $[a,b]$ is of bounded variation, then it is differentiable on $(a,b)$ a.e and $f'$ is integrable over $[a,b]$.

I wonder whether the converse is true, say, if $f$ defined on $[a,b]$ is differentiable on $(a,b)$ a.e and $f'$ is integrable, must $f$ have bounded variation?

In fact, I run into such concrete question: $$ f=x^\alpha \text{sin}\left(\frac{1}{x^\beta}\right) \text{ for $x\in(0,1]$} $$ and $f(0)=0$. The hint says that we can prove that when $\alpha > \beta$, $f$ is of bounded variation by showing $f'$ is integrable.

So here comes my question above, is such argument true?

Answer 1

Rudin, "Real & Complex Analysis", Theorem 7.21: If $f \in L^1[a,b]$ is differentiable at every point, then $f(x) - f(a) = \int_a^x f'(t) dt$.

Now consider $\psi_+(x) = \int_a^x \max(f'(t),0)\; dt$, $\psi_-(x) = \int_a^x \min(f'(t),0)\; dt$. Both are monotonic, hence of bounded variation. It follows that $f(x) = f(a)+\psi_+(x)+\psi_-(x)$ has bounded variation.

Note: To apply Rudin's theorem, $f$ must be differentiable everywhere, not just a.e.

Answer 2

We know, under the conditions in your problem, that the total variation of $f$ is given by $$V_a^b(f)=\int_a^b|f'(x)|dx$$ and since you're given $f'$ integrable then yes: $f$ is of b.v.

Added to the OP's request:

1) Under the condition of your question, in any subinterval $\,[\alpha,\beta]\subset [a,b]\,$, we have that for some $\,c\in (\alpha,\beta)\,,\,f(\beta)-f(\alpha)=f'(c)(\beta-\alpha)$

2) If $\,\mathcal P:=\{a=x_1<x_2<...<x_{n_p}=b\}\,$ is a partition of $\,[a,b]\,$, then the total variation of $f$ on this interval is defined to be $$V_a^b(f):=\sup_{\mathcal P}\sum_{k=1}^{n_p-1}\left|f(x_{i+1}-f(x_i)\right|$$

3) Can you see now from where the integral of $f'$ comes?