Proof Involving Continuity, Irrational Numbers From Elementary Proof Class

snackAtacck

1. The problem statement, all variables and given/known data
Let f be a non-zero continuous function. Prove or disprove that there exists a unique, real number, x, such that the integral from 0 to x of f(s) w.r.t. s = pi.


2. Relevant equations
If any exist, please let me know.

3. The attempt at a solution



I've come to the conclusion that I set up the problem completely wrong, as I did not notice the existence qualifier was unique. Still, to disprove it comes down to showing that there do not exist any real numbers that satisfy the equation, or showing that there are more than one.

The thing is, I'm not entirely sure of what "non-zero" means. I thought it meant that the function cannot literally be a constant 0 for all inputs, but a classmate of mine has insisted that a function is only non-zero if none of its values are zero.

Anyway, I don't see why letting -[(s^2 +1)^2] be the value of the integral and then finding the corresponding functions for f(s) wouldn't work as an example of a function for which that integral will never be equal to pi. We'd have an example for which all x in R would fail to make the integral of f(s) = pi.

I think that I could figure this one out if I only knew what "non-zero" and "continuous" meant.

micromass

A function is nonzero if it is not constant zero. So for example the function 2x is nonzero, although it has a value where it evaluates to zero.

As for your attempt of proof, I think you've got it correct.