(adsbygoogle = window.adsbygoogle || []).push({}); upper semicontinuity problem (Papa Rudin)

By Papa Rudin I do mean Real and Complex Variables by Walter Rudin. This is part of my grad analysis homework...

Let f be an arbitrary complex function on [tex]\mathbb{R}^1[/tex], and define

[tex]\phi(x,\delta)=\sup\{|f(s)-f(t)|:s,t\in(x-\delta,x+\delta)\}[/tex],

[tex]\phi(x)=\inf\{\phi(x,\delta):\delta > 0\}[/tex].

Prove that [tex]\phi[/tex] is upper semicontinuous, that f is continuous at a point x if and only if [tex]\phi(x)=0[/tex].

I can get the rest (I hope) from there.

The working definition of upper semicontinuous (ucs) is: a function f:X-->R^1 is usc if {x:f(x)<a} is an open set for every a in R^1.

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# Homework Help: Need help with Papa Rudin problem

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