There is a theorem in Stein's book 'Real analysis' that gives a sufficient and necessary condition for a curve $\gamma(t) = (x(t), y(t))$ to be rectifiable, which is as follows:
Theorem 3.1(page 116):A curve parametrized by $(x(t), y(t)),\, a ≤ t ≤ b$, is rectifiable if and only if both $x(t)$ and $y(t)$ are of bounded variation (you can check the definition of 'bounded variation function' here).
The proof is straightforward, but I was struggling to gain a little intuition about this theorem, for example letting $x(t) = t$ means the graph of function shouldn't oscillate so much on a small enough domain.
I have 2 questions:
oscilation is the only problem that can happen when dealing with the case $x(t) = t$ ?
can $x(t)$ be of unbounded variation and curve still be graph of a function?
Actually answering the second question answers the first question too. Thanks