I read that if $f,g: X\rightarrow C$ are continuous functions from a topological space to a convex set of an Euclidean space, then the function $F: X\times I\rightarrow C$ defined by $$ F(x,t) = (1-t)f(x) + tg(x) $$ is a straight-line homotopy from $f$ to $g$. May I wonder why is it continuous? I know that I should use the fact that the composition of continuous functions is continuous, and I proved that the function that sends $$ (x,t)\mapsto ((1-t,t),(f(x),g(x))) $$ is continuous, but I was stuck there. I was stuck on this example for about two hours so I really appreciate if someone could show me what the composition functions should look like.
Thanks!
The sum of continuous functions is continuous (by composition with the continuous addition function), so if we can show $(t,x)\to tg(x)$ is continuous it should be clear that this is enough. But the scalar product map $\Bbb R\times\Bbb R^n\to\Bbb R^n$ is continuous and so are its restrictions and corestrictions.
It then suffices to show $(t,x)\to(t,g(x))$ is continuous but that’s really rather clear as it is simply $1\times g$ and the product of continuous functions is continuous.