# Category:Continuity

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This category contains results about continuity, in all its various contexts.

Definitions specific to this category can be found in Definitions/Continuity.

The mapping $f$ is **continuous at (the point) $x$** (with respect to the topologies $\tau_1$ and $\tau_2$) if and only if:

- For every neighborhood $N$ of $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.

## Subcategories

This category has the following 12 subcategories, out of 12 total.

### A

### C

### P

### U

## Pages in category "Continuity"

The following 23 pages are in this category, out of 23 total.

### C

### D

- Definite Integral of Function satisfying Dirichlet Conditions is Continuous
- Definite Integral of Uniformly Convergent Series of Continuous Functions
- Derivative of Uniformly Convergent Sequence of Differentiable Functions
- Derivative of Uniformly Convergent Series of Continuously Differentiable Functions