Γ is a rectifiable simple closed curve and Ω is its interior. f is holomorphic on Ω and continuous on Ω∪Γ. Prove the integral of f along Γ is 0.
Since we already know the Cauchy's integral theorem for a piecewise-smooth curve, I want to use piecewise-linear curve (say, γ) to approximate Γ. But the problem is that I can't assure that γ is contained in Γ∪Ω, so f may be not defined on all of γ. How can I verify it?