explanation of uniform continuity real analysis
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Intuitively uniform continuity means your function can't get infinitely steeper on(a,b) the way 1/x does on(0,1). Steepness doesn't only mean that f may go off to infinity in your interval. It could also oscillate around in some unpleasant way. Look for example at f(x)=sin(1/x) on(0,1).
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