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Let $X$ be a topological space. I want to show that the following map is a homotopy. $h(t,s): [0,1] \times [0,1] \to X$, with: $$h(t,s) = \left\{ \begin{array}{c c} \alpha(\frac{s}{v(t)})& 0 \le s \le v(t), \\ \beta({\frac{s-v(t)}{1-v(t)}})& v(t) \le s \le 1, \end{array}\right.$$ where $\alpha$ and $\beta$ are paths such that $\alpha (1) = \beta(0)$ and $v(t) = (1-t)\lambda + t \mu$ with $\lambda, \mu \in (0,1)$.

This is given as the homotopy which will show that it doesn't matter how I join two paths. That is to say if I concatenate two paths at $\lambda$ to form a new path and I also concatenate them at $\mu$ to form another then these two new maps are homotopy equivalent.

I am struggling to show this is continuous. I thought about taking some point in the pre image of an open set in $X$ and trying to get an open neighbourhood of this point also contained in the pre image.

Trying to do this I thought I could fix $t$ and get a set that looks like $U \times \{t\} \subseteq h^{-1}(X)$ with $U$ open and containing $s$, and I could. But then I realised that what I was doing was just showing it is continuous in each of its variables separately which won't be very helpful.

Thus I am stuck and would appreciate if someone could help me show this is continuous.

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    $\begingroup$ If I break up my domain into two closed pieces and define continuous functions on each piece that agree on the intersection of the pieces, the result is a continuous function. $\endgroup$ – Dylan Wilson Feb 8 '15 at 14:42
  • $\begingroup$ Also called the gluing lemma or pasting lemma. $\endgroup$ – Dan Rust Feb 9 '15 at 13:41

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