Show by giving a concrete example that
$$\int_{\gamma}f(w)dw=\int_{a}^{b} f(z(t))z'(t)dt$$
where $$\int_{\gamma}f(w)dw=\int_{a}^{b} f(t)dz(t),$$
is not always true with weaker assumptions.
(The assumptions that the formula has for the proof is: Let $z:[a,b]\rightarrow \mathbb{C}$ be a function such that $z'$ is continuous, with the right interpretations. Assume also that $f:A\rightarrow \mathbb{C}$ is a continuous function, where $A$ is an open set with $z([a,b]) \subseteq A \subset \mathbb{C}$. Set $\gamma = \{ z(t):t\in [a,b]\}$.)
I was thinking that maby if $z$ has a discontinuous derivative in $[a,b]$ the above formula is not always true. But I'm not sure how to give a concrete example of this.
Can someone please help me? Thanks!