How can we prove that they have at least one intersection point?

Question

Let $f:[0,1]\rightarrow [0,1]$ be a continuous function. The graph of such a function is a continuous curve in the interval $[0,1]$ and it takes values $0\leq y\leq 1$, right?

How can we prove, using the graph, that the graph of $f$ and the line $y=x$ have at least one intersection point?

Answer 1

If $f(0)=0$ or $f(1)=1$, ok. Suppose otherwise and define $g(x)=f(x)-x$. Clearly $g$ is continuous, $f(0)>0$ and $f(1)<1$. Then $g(0)>0>f(1)$; thus, by the intermediate value theorem there exists $x\in[0,1]$ such that $g(x)=0$, i.e., $f(x)=x$.

Answer 2

Use the intermediate value theorem for the function $g(x) = f(x) - x$.