deformation retraction as mapping cylinder

Question

In Hatcher's Algebraic Topology, the mapping cylinder is defined as the quotient space of the disjoint union $(X\times I)\sqcup Y$ (where $I$ is the unit interval) of a continuous $f:X\to Y$, where $(x,1)\in X\times I$ is is identified with $f(x)\in Y$. My question is: need this cylinder be continuous at $(x,1)$? if so can somebody provide a sketch of why? I cant see why $(x,t)$ must be smoothly connected to $f(x)$