Suppose that f is continuous on (0,1) and that int[0,x] f = int[x,1] f

[kbgregory]

Homework Statement

Suppose that f is continuous on (0,1) and that

int[0,x] f = int[x,1] f

for all x in [0,1]. Prove that f(x)=0 for all x in [0,1].

Homework Equations

We know that since f is continuous on (0,1), F(x) = int[0,x] f and F'(x) = f(x) for x in (0,1).

The Attempt at a Solution

What I have so far is:

int[0,x] f = int[x,1] f

int[0,x] f = int[1,0] f - int[0,x] f

F(x) = C - F(x) for some constant C

F'(x) = -F'(x)

f(x) = - f (x)

2 f (x) = 0

f (x)=0 for all x in (0,1).

But this does not show that f(x)=0 for x=0 and x=1, and I am supposed to show that f(x)=0 for all x in the closed interval [0,1].

Any hints on how to do this?

 

[kbgregory]

Ok, I just found out from my professor that there is a typo in the problem, and that f is continuous on [0,1], so I think I can take it from here.