# Homework Help: A analysis problem

1. Feb 20, 2010

### ren_101

1. The problem statement, all variables and given/known data
$$f$$ is continuous on [a,b]
$$f_{1}(x)=\int^x_a f(t)dt$$
$$f_{2}(x)=\int^x_a f_{1}(t)dt$$
.....
$$\forall x\in[a,b],\exists n$$ depends on x , such that $$f_{n}(x)=0.$$
prove that $$f\equiv0.$$

2. Relevant equations
mathematical analysis
3. The attempt at a solution
copy the taylor theorem 's proof???
but I get nothing.

2. Feb 21, 2010

### HallsofIvy

If, for some n, fn is identically 0, then it is a constant and so its derivative is identically equal to 0. But, by the fundamental theorem, the derivative of $f_n(x)= \int_a^x f_{n-1}(t)dt$ is fn-1(x). Therefore, if fn(x) is identically 0 on [a, b], so is fn-1(x).