# Category:Definitions/Continuity

This category contains definitions related to continuity, in all its various contexts.

Related results can be found in Category: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 $f \left({x}\right)$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \left({M}\right) \subseteq N$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### C

### P

## Pages in category "Definitions/Continuity"

The following 96 pages are in this category, out of 96 total.

### C

- Definition:Continuity
- Definition:Continuity/Metric Subspace
- Definition:Continuous at Point of Metric Space
- Definition:Continuous at Point of Topological Space
- Definition:Continuous Extension
- Definition:Continuous Extension/Real Function
- Definition:Continuous Function
- Definition:Continuous Mapping
- Definition:Continuous Mapping (Metric Space)
- Definition:Continuous Mapping (Metric Space)/Point
- Definition:Continuous Mapping (Metric Space)/Point/Definition 1
- Definition:Continuous Mapping (Metric Space)/Point/Definition 2
- Definition:Continuous Mapping (Metric Space)/Point/Definition 3
- Definition:Continuous Mapping (Metric Space)/Point/Definition 4
- Definition:Continuous Mapping (Metric Space)/Space
- Definition:Continuous Mapping (Metric Space)/Space/Definition 1
- Definition:Continuous Mapping (Metric Space)/Space/Definition 2
- Definition:Continuous Mapping (Topological Spaces)
- Definition:Continuous Mapping (Topology)
- Definition:Continuous Mapping (Topology)/Everywhere
- Definition:Continuous Mapping (Topology)/Everywhere/Open Sets
- Definition:Continuous Mapping (Topology)/Everywhere/Pointwise
- Definition:Continuous Mapping (Topology)/Point
- Definition:Continuous Mapping (Topology)/Point/Filters
- Definition:Continuous Mapping (Topology)/Point/Open Sets
- Definition:Continuous Mapping (Topology)/Set
- Definition:Continuous Mapping at Point (Topology)
- Definition:Continuous Mapping Between Topological Spaces at Point
- Definition:Continuous Mapping on Set
- Definition:Continuous on Interval
- Definition:Continuous on Metric Space
- Definition:Continuous Real Function
- Definition:Continuous Real Function at Point
- Definition:Continuous Real Function on Closed Interval
- Definition:Continuous Real Function on Half Open Interval
- Definition:Continuous Real Function on Interval
- Definition:Continuous Real Function on Open Interval
- Definition:Continuous Real Function on Subset
- Definition:Continuous Real Function/Closed Interval
- Definition:Continuous Real Function/Closed Interval/Definition 1
- Definition:Continuous Real Function/Closed Interval/Definition 2
- Definition:Continuous Real Function/Everywhere
- Definition:Continuous Real Function/Half Open Interval
- Definition:Continuous Real Function/Interval
- Definition:Continuous Real Function/Left-Continuous/Point
- Definition:Continuous Real Function/One Side
- Definition:Continuous Real Function/Open Interval
- Definition:Continuous Real Function/Point
- Definition:Continuous Real Function/Right-Continuous/Point
- Definition:Continuous Real Function/Subset
- Definition:Continuous Real-Valued Function
- Definition:Continuous Real-Valued Vector Function

### D

- Definition:Discontinuity
- Definition:Discontinuity of the First Kind
- Definition:Discontinuous Mapping
- Definition:Discontinuous Mapping/Real Function
- Definition:Discontinuous Mapping/Real Function/Point
- Definition:Discontinuous Mapping/Topological Space
- Definition:Discontinuous Mapping/Topological Space/Point

### E

### L

- Definition:Left-Continuous at Point
- Definition:Lipschitz Condition
- Definition:Lipschitz Constant
- Definition:Lipschitz Continuity
- Definition:Lipschitz Continuity/Lipschitz Constant
- Definition:Lipschitz Continuity/Point
- Definition:Lipschitz Continuity/Real Function
- Definition:Lipschitz Continuous at Point
- Definition:Lower Semicontinuous
- Definition:Lower Semicontinuous on Subset
- Definition:Lower Semicontinuous/Subset

### P

### R

- Definition:Real Continuous Extension
- Real Function is Continuous at Isolated Point
- Definition:Removable Discontinuity
- Definition:Removable Discontinuity of Real Function
- Definition:Removable Discontinuity of Real Function/Definition 1
- Definition:Removable Discontinuity of Real Function/Definition 2
- Definition:Right-Continuous at Point