Example of the function $F(x)$ that has non zero total variation as $x$ goes to $-\infty$

I am trying to find an example of the function $F(x)$ that has non zero total variation as $x$ goes to $-\infty$.

Let $T_F(x) := sup\{ \sum_{i=1}^n |F(x_j) - F(x_{j-1})| :n \in \mathbb{N}, -\infty<x_0 < \cdots<x_n=x\}$ and $T_F(-\infty) = \lim_{x \rightarrow -\infty} T_F(x)$.

I considered $F(x) = sin x^2$ as a potential candidate,but I have not yet proven it.

Since $sin x^2$ oscillates more frequently as $x$ goes to $-\infty$, I believe that the absolute value of total variation cannot be zero.

How could I mathematically prove this idea?