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 $g(x)=f(x)-x$.


What is the sign of $g(0)$?
What is the sign of $g(1)$?
What does the intermediate value theorem tell you?
Topology does assume that you know some basic set theory, among which the definition of functions.

If $X,Y$ are sets a function $f:X \to Y$ is just a set of pairs $f = \{x,y): x \in X. y \in Y\}$, such that whenever $(x,y) \in f$ and $(x,y') \in f$ we have $y=y'$ and for all $x \in x, \exists y \in Y: (x,y) \in f$ (every $x \in X$ has a unique image in $Y$, the unique $y$ for $x$ such that $(x,y) \in f$ is usually written $f(x) \in Y$.

Here we are in the situation that $X=Y=[0,1]$ and a fixed point, is an $x \in [0,1]$ such that $f(x) = x$.