Combining two homotopies in linear time Suppose $f,g,h:M\to N$ are smooth maps such that $f\sim g$ and $g\sim h$. I.e. there are smooth homotopies
$$F:M\times I\to N,\qquad G:M\times I\to N,$$
such that
$$F(x,0)=f, F(x,1)=g,\qquad G(x,0)=g, G(x,1)=h$$
I want to know what is the problem with the following homotopy between $f$ and $h$? Isn't smooth?
$$H(x,t)=\begin{cases}F(x,2t), & t\in[0,\frac{1}{2}]\\ G(x,2t-1), & t\in[\frac{1}{2},1]\end{cases}$$
I am asking this in regard to Milnor construction of $f\sim h$ using bump function in Topology from differentiable viewpoint.

This is not smooth at $1/2$. Take for instance $M = \{0\}$ and $N = \Bbb R$, with $F(0,t) = t$ and $G(0, t) = 1-t$. The graph of your $H$
One can overcome "cusps" like this as Milnor does by slowing down to a stop at the cusp-point before moving on to the next part of the curve.
