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Originally Posted by ewmayer
Also, it makes no sense to speak of a function being continuous at a single point - continuity only makes sense in regions, i.e. neighborhoods of a point.
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Although we usually think of it this way, continuity is classically defined at a point. The classic counter-intuitive case is
f(x) = 1/q if x is rational p/q in lowest form,
f(x)= 0 if x is irrational
This function is continuous at irrational points.
(f(x) is continuous at x' if, for every epsilon > 0 there exists a delta such that
for every x such that |x-x'|<delta, |f(x)-f(x')|<epsilon)
William