Continuous Function

Last updated Nov 1, 2022

Let $(X, \tau), (Y, \rho)$ be Topological Spaces and let $f: X \to Y$ be a Function. Then $f$ is a Continuous Function if $\forall V \in \rho$ $$f^{-1}(V) \in \tau$$

1. Sometimes we refer to the Set of Continuous Functions as $C(X, Y)$. If $Y$ is omitted, it is usually understood to be $\mathbb{R}$.
2. We may also refer to the Set as $C^{0}(X)$ if we are working in the context of Differentiation

1. A Function is Continuous iff it preserves Net Convergence
2. A Function is Continuous iff it takes Basis Sets back to Open Sets
3. A Function is Continuous iff it takes Closed Sets back to Closed Sets
4. Continuous Functions Preserve Connectedness
5. Continuous Functions Preserve Compactness
6. Continuous Functions Preserve Limit Points
7. Continuous Functions Preserve Sequence Limits
8. Continuous Functions to Hausdorff Spaces are Determined on Dense Sets

Let $(X, d_{X})$, $(Y, d_{Y})$ be Metric Spaces. Then a Function $f: X \to Y$ is a Continuous Function if for all convergent sequences $x_{n} \to x$ in $X$, $f(x_{n}) \to f(x)$.

# TODO

• Connect together all the different criteria defining Continuous Functions over
  • Metric Spaces
  • Hausdorff, First Countable Spaces

• Function Preimage