Definition 1.19

Definition 1.19: If $$(X,d_X)$$ and $$(Y,d_Y)$$ are metric spaces, we say a function $$f:X \rightarrow Y$$ is continuous at a point $$x$$ if given any $$\epsilon > 0$$ there is some $$\delta > 0$$ so that

$$d_Y(f(x),f(y)) < \epsilon$$ whenever $$d_X(x,y) < \delta$$

A function which is continuous at every point in its domain is said to be a continuous function.