Continuity

continuity, epsilon-delta definition

Let $f : A \to R$ where $A \subseteq R$ and $c \in A$ . Then $f$ is continuous at $c$ if $\forall ε > 0 \exists δ > 0 \forall x \in A (| x - c | < δ \to | f (x) - f (c) | < ε)$

Compare and contrast this with the definition of a Functional Limit.

Here are some remarks:

If $c$ is Isolated Point then $f$ is automatically continuous at $c$ .
If $c$ is a Limit Point of $A$ then this can be phrased in terms of limits.