Proving fixed points of a continuous function **Prove that every continuous function $f:[0,1] \rightarrow [0,1]$ has a fixed point.** What does the statement $f:[0,1] \rightarrow [0,1]$ mean? I have never studied function rigorously so I am not well-versed with the notation. Can somebody help me out or provide some resources that may prove beneficial?

It means that the **domain** of the function is $[0,1]$ and that the **codomain** of the function is $[0,1]$. An example of such a function would be $x^2$, for example, or maybe $\sin(x)$ or an infinitely many other functions.

To solve your problem, here's a couple guidelines:

Take a look at the function $g(x)=f(x)-x$.

1. What is the sign of $g(0)$?
2. What is the sign of $g(1)$?
3. What does the intermediate value theorem tell you?