# 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:

## Subcategories

This category has the following 15 subcategories, out of 15 total.

### A

### C

### P

### U

## Pages in category "Continuity"

The following 27 pages are in this category, out of 27 total.

### C

- Combination Theorem for Continuous Functions
- Composite of Continuous Mappings between Normed Vector Spaces is Continuous
- Composite of Continuous Mappings is Continuous/Corollary
- Continuity of Heaviside Step Function
- Continuous Function on Compact Subspace of Euclidean Space is Bounded
- Continuous Inverse Theorem
- Continuous Real Function Differentiable on Borel Set
- Continuous Replicative Function/Historical Note
- Continuously Differentiable Real Function at Removable Singularity
- Continuously Differentiable Real Function at Removable Singularity/Corollary

### 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