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Suppose a mapping f: R → R satisfies f(x + y) = f(x) + f(y) for every pair of real numbers x and y. If f is continuous at x = 0, which of the following must be true?

1. f(x) = 0 for every real x
2. f(x) is continuous at every positive real number
3. f(x) is continuous for all real x
4. None of the above

Correct answer: f(x) is continuous for all real x

Solution

An additive function f(x+y)=f(x)+f(y) that is continuous at a single point (x=0) is continuous on all of R (in fact f(x)=cx). So f is continuous for all real x.

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