# Continuous Functions

## Key Questions

• You can prove that a function $f \left(x\right)$ is continuous at $a$ by verifying that ${\lim}_{x \to a} f \left(x\right) = f \left(a\right)$.

• No, it is not continuous at $x = 3$ since it is not defined there (zero denominator).

We may also state two alternative definitions of continuous functions, using either the sequential criterion or else using topology and open sets.

#### Explanation:

Alternative definition number 1
Let $f : X \to Y$ be a function and let $\left({x}_{n}\right)$ be a sequence in X converging to an element x in X, ie $\lim \left({x}_{n}\right) = x \in X$
Then f is continuous at x iff and only if the sequence of function values converge to the image of x undr f, ie $\iff \lim \left(f \left({x}_{n}\right)\right) = f \left(x\right) \in Y$

Alternative definition number 2
Let $f : X \to Y$ be a function. Then f is continuous if the inverse image maps open subsets of Y into open subsets in X.
ie, $\forall {A}_{o p e n} \subseteq Y \implies {f}^{- 1} \left(A\right)$ is open in X