$\sup_{\Gamma}\sum |\phi(x_i)-\phi(x_{i-1})|=\infty$ but $\lim_{|\Gamma|\to 0}\sum (\phi(x_i)-\phi(x_{i-1}))$ exists

Question

Let $\Gamma=\{a=x_0,\cdots,b=x_m\}$ a partition of $[a,b]$, is it possible to find a function $\phi$ (any) such that $$\sup_{\Gamma}=\sum_{i=1}^m |\phi(x_i)-\phi(x_{i-1})|=\infty$$ but such that the following limit exists $$\lim_{|\Gamma|\to 0}\sum_{i=1}^m (\phi(x_i)-\phi(x_{i-1}))\ \ \ \ ?$$ I think the answer is yes, I just could not think of the $\phi$ that works.

Answer 1

Note that $\phi(x) = \begin{cases} x \sin \frac{1}{x}, & 0 < x \leqslant 1 \\0, &x = 0\end{cases}$ has unbounded variation but is continuous, and for any partition of $[0,1]$,

$$\sum_{i=1}^m (\phi(x_i)-\phi(x_{i-1})) = \phi(1) - \phi(0) = \sin 1$$