Prove there is no the continous function $g:[a,b]\to \{a,b \}$ such that $g(a)=a, g(b)=b$.

Question

Prove there is no the continous function $g:[a,b]\to \{a,b \}$ such that $g(a)=a, g(b)=b$.

Using properties of connected and proposition that any continuous function $f:[a,b]\to [a.b]$ has fixed point.

Can you help me? Any hint will be appreacite.

Answer 1

The image of a connected set by a continuous map is connected. $[a,b]$ is connected, but not is $\{a,b\}$.

Answer 2

Another possibility: Let $s:\{a,b\} \to [a,b]$ be given by $s(a) = b, s(b)=a$. Then $s$ is continuous and if $g$ existed, $s\circ g$ would be a fixed point free function on $[a,b]$. Contradiction.