Calculus – Continuous Function on $[0,1]$ Not of Bounded Variation

Question

I'm looking for a continuous function $f$ defined on the compact interval $[0,1]$ which
is not of bounded variation.

I think such function might exist. Any idea?

Of course the function $f$ such that
$$
f(x) =
\begin{cases}
1 & \text{if $x \in [0,1] \cap \mathbb{Q}$} \\\\
0 & \text{if $x \notin [0,1] \cap \mathbb{Q}$}
\end{cases}
$$
is not of bounded variation on $[0,1]$, but it is not continuous on $[0,1]$.

Answer 1

Consider any continuous function passing through the points $(\frac1{2n},\frac1n)$ and $(\frac1{2n+1},0)$, e.g. composed of linear segments. It must have infinite variation because $\sum\frac1n=\infty$.