Suppose f is a continuous function on some open subset S of the real numbers, to the positive real numbers. Suppose the integral of f over S is finite.
An open subset of the real line is a union of disjoint open intervals (which could have endpoints at infinity.) Define the length of the open subset as the sum of the length of the intervals.
So the question is, show that for any epsilon > 0, there's a delta such that for any open subset T of S with length(T) < delta, the integral of f over T is less than epsilon.
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(a,b) U [b,c) is an open subset, too. Are you defining a subclass of open intervals with which this particular problem deals, eliminating half-open intervals?
An open subset of the real line is a union of disjoint open intervals. Could be finitely many, could be a union of countably many disjoint open intervals.
A half-open interval is not an open subset of the real line. "Open" means that for each point x in the subset, the interval (x-a,x+a) is contained in the subset, for some nonzero a.
Do you have in mind that all open subsets may be expressed as the union of disjoint open intervals? But that a particular open subset may also be expressed as, say, (a,b) U [b,c)?
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