# Category:Continuous Mappings

This category contains results about **Continuous Mappings**.

Definitions specific to this category can be found in **Definitions/Continuous Mappings**.

The concept of **continuity** makes precise the intuitive notion that a function has no "jumps" at a given point.

Loosely speaking, in the case of a real function, **continuity at a point** is defined as the property that the graph of the function does not have a "break" at the point.

Thus, a small change in the independent variable causes a similar small change in the dependent variable

This concept appears throughout mathematics and correspondingly has many variations and generalizations.

## Subcategories

This category has the following 26 subcategories, out of 26 total.

## Pages in category "Continuous Mappings"

The following 109 pages are in this category, out of 109 total.

### C

- Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous
- Characterization of Continuity in terms of Nets
- Closed Image of Closure of Set under Continuous Mapping equals Closure of Image
- Closure of Image under Continuous Mapping is not necessarily Image of Closure
- Closure of Preimage under Continuous Mapping is not necessarily Preimage of Closure
- Combination Theorem for Continuous Mappings
- Compactness is Preserved under Continuous Surjection
- Compactness Properties Preserved under Continuous Surjection
- Compactness Properties Preserved under Projection Mapping
- Complex-Differentiable Function is Continuous
- Composite of Continuous Mappings at Point is Continuous
- Composite of Continuous Mappings on Metric Spaces is Continuous
- Constant Function is Continuous
- Constant Mapping is Continuous
- Continuity Defined by Closure
- Continuity Defined from Closed Sets
- Continuity in Initial Topology
- Continuity of Composite with Inclusion
- Continuity of Linear Transformation/Normed Vector Space
- Continuity of Mapping between Metric Spaces by Closed Sets
- Continuity Test for Real-Valued Functions
- Continuity Test using Basis
- Continuity Test using Sub-Basis
- Continuous Bijection from Compact to Hausdorff is Homeomorphism
- Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary
- Continuous Image of Connected Space is Connected/Corollary 2
- Continuous Involution is Homeomorphism
- Continuous Linear Transformations form Subspace of Linear Transformations
- Definition:Continuous Map Induced by Continuous Mapping
- Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness
- Continuous Mapping is Continuous on Induced Topological Spaces
- Continuous Mapping is Measurable
- Continuous Mapping is Sequentially Continuous
- Continuous Mapping is Sequentially Continuous/Corollary
- Continuous Mapping of Separation
- Continuous Mapping on Finite Union of Closed Sets
- Continuous Mapping on Union of Open Sets
- Continuous Mapping to Product Space
- Continuous Mapping to Product Space/Corollary
- Continuous Mapping to Product Space/General Result
- Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide
- Continuous Mappings preserve Convergent Sequences
- Countability Axioms Preserved under Open Continuous Surjection
- Countability Properties Preserved under Projection Mapping
- Countable Compactness is Preserved under Continuous Surjection

### D

- Derivative Operator on Continuously Differentiable Function Space with C^1 Norm is Continuous
- Derivative Operator on Continuously Differentiable Function Space with Supremum Norm is not Continuous
- Diagonal Operator over 2-Sequence Space is Continuous Linear Transformation
- Distance from Point to Subset is Continuous Function
- Distance Function of Metric Space is Continuous
- Domain of Continuous Injection to Hausdorff Space is Hausdorff

### E

### F

### G

### I

### L

- Left Shift Operator on 2-Sequence Space is Continuous
- User:Leigh.Samphier/Topology/Continuous Mapping Induced by Continuous Map is Continuous
- Lindelöf Property is Preserved under Continuous Surjection
- Linear Integral Bounded Operator is Continuous
- Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous
- Local Connectedness is not Preserved under Continuous Mapping
- Definition:Localic Mapping Induced by Continuous Mapping

### M

- Mapping from Cartesian Product under Chebyshev Distance to Real Number Line is Continuous
- Mapping from L1 Space to Real Number Space is Continuous
- Mapping from Standard Discrete Metric on Real Number Line is Continuous
- Mapping whose Graph is Closed in Chebyshev Product is not necessarily Continuous
- Maximum Rule for Continuous Functions
- Metric is Continous Mapping
- Minimum Rule for Continuous Functions
- Multiple Rule for Continuous Mapping to Normed Division Ring

### P

- Paracompactness is not always Preserved under Open Continuous Mapping
- Paracompactness is Preserved under Projections
- Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets
- Projection from Product Topology is Continuous
- Projection from Product Topology is Open and Continuous

### R

### S

- Second-Countability is Preserved under Open Continuous Surjection
- Sequential Compactness is Preserved under Continuous Surjection
- Sigma-Compactness is Preserved under Continuous Surjection
- Supremum Operator Norm as Universal Upper Bound
- Supremum Operator Norm is Norm
- Supremum Operator Norm is Well-Defined