Continuous function on a metric space/topology X a metric space. Let $\gamma$1, $\gamma$2 : [0,1] → X be two continuous paths such that $\gamma$1 (1) = $\gamma$2 (0). Consider the map$\gamma$ : [0,1] → X deﬁned as $\gamma$(t) =  $\gamma$1 (2t), if t∈ [0,$\frac{1}{2}$)
or $\gamma$(t) = $\gamma$2 (2t−1), if t ∈ [$\frac{1}{2}$,1]  Show that $\gamma$ is a continuous path in X.
How should I show if this is continuous? 
Can I use the "usual" definitions, as the epsilon-delta or the limit definition? Or do I have to use something else since X is a metric space?
Any guidence or help would be greatly appreciated! 
 A: It is also true more generally if $X$ is a topological space not necessarily metrizable.

In general if $A,B$ are closed subsets of topological space $Y$ and we have function $f:A\to X$ continuous on $A$, function $g:B\to X$ continuous on $B$ such that $f$ and $g$ coincide on $A\cap B$ then the function $h=f\cup g:A\cup B\to X$ is continuous as well.
To prove that $h$ is continuous it is enough to show that the preimage of a closed set  $F\subseteq X$ is a closed subset of $Y$. 
This preimage is the set $f^{-1}(F)\cup g^{-1}(F)$, so we are ready if we can prove that the sets $f^{-1}(F)$ and $g^{-1}(F)$ are closed. 
Since $f$ is continuous the set $f^{-1}(F)$ is a closed set in space $A$ which means that it can be written as $P\cap A$ where $P$ is a closed set in $Y$. 
But that means that $f^{-1}(F)=P\cap A$ is an intersection of two closed sets: $A$ and $P$. 
We conclude that $f^{-1}(F)$ is closed.
Similarly we find that $g^{-1}(F)$ is closed and we are ready.

You can apply that here on $A=[0,0.5]$ and $B=[0.5,1]$ wich are closed subsets of $Y:=[0,1]$.
