I'm trying to prove a result similar to Runge's theorem and Mergelyan's theorem (link at the bottom of the previous link), but without the condition of analyticity. The problem is as follows:
Let γ : [0,1] → $\mathbb{C}$ be a C$^1$ curve and K a compact subset disjoint from Γ = {γ(t) : t ∈ [0,1]}. Let f : Γ→ $\mathbb{C}$ be continuous. Define g : K → $\mathbb{C}$ by g(z)=$\int_{\gamma} \frac{f(w)}{w-z} dw$. By splitting up the interval [0, 1] into subintervals, prove g can be uniformly approximated on K by functions of the form $g_n(z)=\sum_{j=1}^{n} \frac{A_j}{w_j - z}$, where $w_j$ ∈ Γ and A$_j$ ∈ $\mathbb{C}$ for each j = 1, 2, ..., n.
The problem is, we're trying to approximate g which, unless I'm wrong, doesn't appear to be analytic. As the question suggests, I suppose we're essentially trying to split the integral over [0,1] into n subintervals, each of which will correspond to a term in the sum g$_n$, and then simply apply the triangle inequality to show the integral is uniformly approximated by the g$_n$, or something along those lines. However, I don't know how we should split the integral; I feel like I should be using something like Runge's theorem because otherwise I don't see what all the 'compact, disjoint' setup is for, but as I've said, I don't know how to apply it: I suppose presumably f is uniformly continuous since K is compact; does this give us a starting point for where to split up [0,1]? Any guidance would be much appreciated - it's personal revision rather than homework, but I'll mark it as homework anyway. Thank-you!