How to show a straight line homotopy is continuous?

Question

Given $f$ and $g$ continuous maps from $X$ into $\mathbb{R}^{2}$, how to show that the straight line homotopy map $F(x,t)=(1-t)f(x)+tg(x)$ is continuous?

Answer 1

If $M:\mathbb R^3\to \mathbb R^2$ is the map given by $(x,y,t)\mapsto (tx,ty)$ and $A:\mathbb {R^2\times R^2}\to \mathbb R^2$ is the map given by $((x_1,y_1),(x_2,y_2))\mapsto (x_1+x_2,y_1+y_2)$ then $M$ and $A$ are continuous maps. Also the maps $I:\mathbb {R\to R}$ given by $t\mapsto t$ and $S:\mathbb{\to R}$ given by $t\mapsto (1-t)$ are continuous.

Then you can see that your map $F$ is given by $$(x,t)\mapsto((x,t),(x,t))\mapsto(f(x),(1-t),g(x),t)\mapsto((1-t)f(x),tg(x))\mapsto(1-t)f(x)+tg(x)$$

Where the second map is $f\times S\times g\times I$, the thirsd map is $M\times M$ and the fourth map is $A$. I hope you are convinced that the first map is continuous.

Answer 2

If both $f$ and $g$ are continuous then $F$ is a composition of continuous functions which is also continuous.