Linear homotopy Let $\lambda, \mu:[a,b]\longrightarrow X\subset\mathbb{R}^n$ paths such that the straight line $[\lambda(s),\mu(s)]$ lies in X for all $s\in[a,b]$. Set:
$$\begin{array}{lccc}
H:&[a,b]\times[0,1]&\longrightarrow&X\\
&(s,t)&\longrightarrow&(1-t)\lambda(s)+t\mu(s)
\end{array}$$
How can I show that $H$ is continuous?

Since $\lambda$ and $\mu$ are paths, by the definition, they are continuous. $1-t$ and $t$ are clearly continuous.
So we know that the sum and product of continuous functions are continuous, therefore $H$ is continuous.
