Definition:Continuous Mapping (Topology)/Everywhere

Definition
Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces.

Let $f : A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

The mapping $f$ is continuous everywhere (or simply continuous) iff $f$ is continuous at every point $x \in A_1$.

Definition by Open Sets
Equivalently, continuity of $f$ can be defined using open sets:

That these conditions are equivalent is proven in Continuous Mapping by Open Sets.

Also see

 * Continuous Mapping by Open Sets