I tried solving the problems from Munkres (Topology):
Given spaces $X,Y$ let $[X,Y]$ denotes the homotopy classes of maps of $X$ into $Y.$
(a) Let $Y=[0,1],$ show that for any set $X,$ the set $[X,Y]$ has a single element.
(b) If $Y$ is path connected and $X=[0,1]$, show that $[X,Y]$ has a single element.
So precisely for both the cases I am expected to show any two continuous maps $f,g:X\to Y$ are homotopic.
For (a) I have defined the homotopy as $G(x,t)=(1-t)f(x)+t.g(x)$ and for (b) I have defined the homotopy as $F(x,t)=p_x(t)$ where $p_x$ is the path joinning $f(x)$ and $g(x).$
Unfortunately I could not show $G$ and $F$ are continuous. Please help.