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Suppose f: R → R satisfies f((x+y)/3) = (f(x)+f(y))/3 for all real x,y, along with f(0)=0 and f'(0)=3. Which of the following must be true?

1. f(x) is a quadratic function
2. f(x) is continuous but not differentiable
3. f(x) is differentiable on R
4. f(x) remains bounded on R

Correct answer: f(x) is differentiable on R

Solution

The functional equation given implies that f is a linear function, and since it is differentiable at 0 with a defined derivative, it must be differentiable everywhere on R.

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