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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!

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Apparently I can't comment on your post so I'll have to post as an answer.

$g$ is an analytic function in $\mathbb{C}\setminus \Gamma$, in particular it is analytic in $K$ so Runge's theorem applies. The problem is the specific approximating sequence you're asked to derive and I don't know right now how to handle it.

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    Ah, well now I know g is analytic I have a few more relevant lemmas at my disposal - though first, could you please explain how we justify that g is analytic in that region? Lemmas which are relevant: 1. K $\subset \mathbb{C}$ compact, and $\Omega$ open and containing K. Then $\exists$ a finite number of piecewise linear closed curves $\Gamma_1,...,\Gamma_m \subset \Omega - K$ such that for every analytic f: $\Omega \to \mathbb{C}$, and every z $\in$ K, f(z)=$\sum_{j=1}^{m} \frac{1}{2 \pi i} \int_{\Gamma_j} \frac{f(w)}{w-z} dw$. I have another handy lemma, but it may require another comment.2011-01-22
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    Second lemma: K$\subseteq \mathbb{C}$ compact, $\Omega \subseteq \mathbb{C}$ open, containing K, and if f:$\Omega \to \mathbb{C}$ is analytic and $\epsilon$>0, then $\exists$ points $\zeta_1,...,\zeta_p \in \Omega - K$ and $A_1,...,A_p \in \mathbb{C}$ such that $|f(z)-\sum_{j=1}^n \frac{A_i}{z-\zeta_i}| < \epsilon$. This seems like it should be relevant, but we need w$_j$ in $\Gamma$, and I don\t see how points we end up choosing are necessarily even in the neighbourhood of $\Gamma$. The 'proof' I was given was "approximate integrals by suitable Riemann sums", so seems likely to be relevant...2011-01-22
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    "Approximate integrals by suitable Riemann sums" should be what you need to solve your problem ...2011-10-30