Category:Continuous Mappings
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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 28 subcategories, out of 28 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
- User:Leigh.Samphier/Topology/Continuous Mapping of Subspace into Hausdorff Space has At Most One Extension to Closure
- 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