Suppose $f:[0,1]\to\Re$ and $g:[0,1]\to\Re$ are continuous, and their images both coincide with the closed unit interval: $f([0,1])=g([0,1])=[0,1]$. Prove that there exists ${x_0}\in[0,1]$ such that $f(x_0)=g(x_0)$.
So, I know I am going to be using the intermediate value theorem, but I am confused on how to get some actual values since there is technically no $f(x)$ or $g(x)$ defined. I did what I usually do in an IVT problem and I let $y(x)=f(x)-g(x)=0$. From here, I don't know which values to plug in to check if they are in between $[0,1]$ since I only have information about their image, continuity, and domain.
My professor suggested using a smaller interval between the minimizer and the maximizer of the function $f(x)$. But, how would I go about finding a minimizer and maximizer of a function that is not explicitly defined?