# Let f:[a,b]->[a,b] be a continuous function. Prove that f has a fixed point, i.e. that there is a c in [a,b] such that f(c)=c

(d) Let $f:\left[a,b\right]\to \left[a,b\right]$ be a continuous function. Prove that $f$ has a fixed point, i.e. that there is a $с\in \left[a,b\right]$ such that $f\left(c\right)=c$.
In the solution, it says that $f\left(a\right)\ge a$ and $f\left(b\right)\le b$ but it do not seem obvious for me. If I am just given that a $f:\left[a,b\right]\to \left[a,b\right]$, how do I know is this function increasing, decreasing or just a horizontal line?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

kartonaun
Recall what $f:\left[a,b\right]\to \left[a,b\right]$ means. It means that given any $x\in \left[a,b\right]$, we know $f\left(x\right)\in \left[a,b\right]$. What does it mean for $f\left(x\right)$ to be in the interval $\left[a,b\right]$? That $a\le f\left(x\right)\le b$. For $a$, $a\le f\left(a\right)\le b$ and for $b$, $a\le f\left(b\right)\le b$. See that nothing about the shape or behavior of $f$ is requiblack: just that the range, or image of $f$ is a subset of $\left[a,b\right]$.