Continuity/more general pasting lemma for compactly generated Hausdorff spaces I constructed a map and would like it to be continuous for nice spaces, but I was having a little trouble with point-set topology... A simplified version is as follows.
Let $(X,x_0)$, $(Y,y_0)$ be pointed topological spaces. Take any continuous maps
\begin{eqnarray}
&&h: X \rightarrow I \\
&&f: X \times I \rightarrow Y
\end{eqnarray}
such that
\begin{eqnarray}
&&h^{-1}(0) = \{x_0\} \\
&& f(x_0,t) = y_0~~\forall t\\
&&f(x,0)=f(x,1)=y_0 ~~\forall x
\end{eqnarray}
(Equivalently, $f:(X,x_0) \rightarrow (\Omega Y,c_{y_0})$ with the latter given the compact-open topology.) Define
\begin{eqnarray}
F: X \times I \times I \rightarrow Y 
\end{eqnarray}
by
\begin{eqnarray}
F(x,t,s) = \begin{cases}
f(x, \frac{t}{h(x)+s}), & t< h(x) + s \\
y_0, & t \geq h(x)+s
\end{cases}
\end{eqnarray}
Now, $F$ should not be continuous in general, but if we work in the category of compactly generated Hausdorff spaces (i.e. $X$, $Y$ are CGH and all products are compactly generated products), is $F$ always continuous? If not, is there a category where it is?

The map $F$ is indeed continuous in general. The following prove is inspired by lemma $5.4.3$ in tom Dieck's Algebraic Topology.
Assume for the moment that $F$ is a function from $X×[0,2]×I$ to $Y$ defined by the rule given in your post. Let $C = \{ (x,t,s) \mid t\le h(x)+s \}$ be a subspace of $X×[0,2]×I$ and let 
$$
\begin{alignat}{2}
q:\; & X\times I\times I && \to C \\
& (x,t,s) && \mapsto (x,t(h(x)+s),s)
\end{alignat}
$$
This map is easily seen to be surjective. Let us show that $q$ is a quotient map: The map 
$$
\begin{alignat}{2}
\Gamma:\; & X×I×I && \to X×I×I×I,\\
& (x,t,s) && \mapsto (x,t,h(x),s) \\
\end{alignat}
$$ 
is an embedding onto a closed subspace $D$. Since the map 
$$
\begin{alignat}{2}
m:\; & I×I×I && \to [0,2]×I, \\
& (a,b,c) && \mapsto (a(b+c),c)
\end{alignat}
$$
is perfect, $M = \text{id}_X×m$ is closed. The restriction of $M$ to $D$ is closed, hence $M\Gamma = q$ is closed and therefore a quotient map.
Now $Fq = fd_s$ where $d_s(x,t,s)=(x,t)$. That means $Fq$ is continuous which implies that $F$ is continuous on $C$. On $\{(x,t,s)\mid t\ge h(x)+s\}$, $F$ is clearly continuous, hence it is continuous on its entire domain. Now we simply restrict this map to $X×I×I$ and we are finished.
