Suppose that $f$ is a continuous real-valued function on $[a, b]$ with $f(a)<f(b)$. If $f(a)<v<f(b)$ show that there exists $c$ with $a<c<b$ and $f(c)=v$.

Deduce that if $f$ is a continuous function from the closed bounded interval $[a, b]$ to itself, there exists at least one fixed point, i.e., a number $d$ belonging to $[a, b]$ with $f(d)=d$. Does this fixed point property remain true if $f$ is a continuous function defined (i) on the open interval $(a, b)$ and (ii) on $\mathbb{R}$ ? Justify your answers.