- #1

snipez90

- 1,101

- 5

## Homework Statement

Find all continuous functions f: R -> R such that for all real x and all natural numbers n we have

(n^2)*integral[x to x + 1/n of f(t)dt] = n*f(x) + 1/2

## Homework Equations

FTC (already applied)

Mean Value Theorem

## The Attempt at a Solution

Suppose a solution exists. Note that the LHS is differentiable according to the FTC. Hence, the RHS is differentiable (in particular, f is differentiable). We can rewrite the LHS as n^2{ integral[0 to x + 1/n of f(t)dt] - integral[0 to x of f(t)dt] }. Then differentiating both sides of the equation gives n^2[f(x + 1/n) - f(x)]= n*f'(x) or n[f(x + 1/n) - f(x)]= f'(x) for all x and all n. But this is the same as saying that there exists an x in (x, x+1/n) such that f'(x) = [f(x + 1/n) - f(x)] / (1/n) by the Mean Value Theorem.

Unfortunately, I am not sure how to proceed from here. It seems like the "calculus portion" is done but I can't see what kind of analysis I should be doing to finish the problem.