Show that if $X$ is path-connected and $f:X\to Y$ is a continuous map, then the image $f(X)$ is path-connected.
In order to show this is path connected I know the definition is :
Definition: A topological space $X$ is path connected if $\forall x,y \in X \, \, \exists$ continuous function $\gamma: [0,1] \rightarrow X$ such that $\gamma(0)=x$ and $\gamma(1) = y$. i.e. any two points can be connected with a continuous path.