# 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 19 subcategories, out of 19 total.

### A

- Absolutely Continuous Functions (10 P)

### C

- Continuous Geometry (empty)
- Continuous Operators (1 P)

### D

### L

- Lipschitz Continuity (1 P)

### P

### S

### U

- Uniform Continuity (3 P)

## Pages in category "Continuity"

The following 28 pages are in this category, out of 28 total.

### C

- Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
- 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 Discontinuity
- Continuously Differentiable Real Function at Removable Discontinuity/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
- Distribution acting on Sequence of Test Functions without common Support is not Continuous