I'm familiar with the Real Analysis proof of the chain rule (i.e. looking at the difference quotient for both g(f(z)) and for f(z)), and I'm familiar with another proof using the Weierstrass definition of differentiability (differentiable iff there is a continuous function such that ...).
But in Bak and Newman's Complex Analysis they give a hint for proving that the composition of differentiable functions is differentiable.
Begin by noting g(f(z+h))−g(f(z))=[g′(f(z))+ϵ][f(z+h)−f(z)]
where ϵ→0 as h→0.
This seems to me to be practically assuming the thing we're trying to prove. What is the justification for this equation? It's not an equation I've encountered in earlier studies--am I supposed to be familiar with it?
This also isn't the first time that I've encountered an expression involving quantities going to 0 like this, which I didn't fully understand (like when reading about Machine Learning or Statistics). Is there a book I can consult to better understand the theory around this? As far as I recall it wasn't in baby Rudin or Ross's Analysis textbook, and yet it comes up kind of often and is thrown around casually.
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