In Tao's Notes 3 on complex analysis there is the following Exercise (number 10)
Find a proof of Goursat's theorem that avoids explicit use of proof by contradiction.
Goursat's theorem states that $U \subset \mathbb{C}$ is open, $f: U \to \mathbb{C}$ is holomorphic and $\Delta \subset U$ is a solid triangle then the complex line integral $$\int_{\partial \Delta} f(z) \,\mathrm{d} z$$ vanishes.
A Hint ist given:
Use the fact that a solid triangle is compact, in the sense that every open cover has a finite subcover.
I can find multiple proofs without explicit use of contradiction if I assume that the derivative $f'$ is continuous (that is, if $\forall \varepsilon > 0: |x| < \varepsilon \Rightarrow x = 0$ doesn't use contradiction).
Any ideas on how to prove it without using the continuity of the derivative?