# Category:Continuous Mappings

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This category contains results about continuous mappings in the context of Topology.

## Subcategories

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

## Pages in category "Continuous Mappings"

The following 98 pages are in this category, out of 98 total.

### C

- Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous
- 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 is Continuous
- Composite of Continuous Mappings is Continuous/Point
- 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 from Union of Restrictions
- Continuity of Composite with Inclusion
- Continuity of Composite with Inclusion/Inclusion on Mapping
- Continuity of Composite with Inclusion/Mapping on Inclusion
- Continuity of Composite with Inclusion/Uniqueness of Induced Topology
- Continuity of Linear Transformations/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
- Continuous Image of Connected Space is Connected/Corollary 2
- Continuous Involution is Homeomorphism
- 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 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
- 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

### G

### I

### L

### 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