(d) Let $f:[a,b]\to [a,b]$ be a continuous function. Prove that $f$ has a fixed point, i.e. that there is a $\u0441\in [a,b]$ such that $f(c)=c$.

In the solution, it says that $f(a)\ge a$ and $f(b)\le b$ but it do not seem obvious for me. If I am just given that a $f:[a,b]\to [a,b]$, how do I know is this function increasing, decreasing or just a horizontal line?

In the solution, it says that $f(a)\ge a$ and $f(b)\le b$ but it do not seem obvious for me. If I am just given that a $f:[a,b]\to [a,b]$, how do I know is this function increasing, decreasing or just a horizontal line?