All Relationships between Continuous Functions of the Real Numbers and Infinite Sets

1. Fundamental Properties:
– Every continuous function on a closed interval [a,b] maps to a closed interval [m,M] (Extreme Value Theorem)
– Continuous functions on the reals map connected sets to connected sets
– The image of a compact set under a continuous function is compact
– Every continuous function on R is measurable (Lebesgue measurable)
– A continuous function is uniformly continuous on any compact subset of its domain

2. Cardinality Relationships:
– The set of all continuous functions from R to R is uncountably infinite
– The set of real numbers R is uncountably infinite
– Any non-degenerate interval of real numbers is uncountably infinite
– The graph of a continuous function contains uncountably many points
– The set of discontinuities of any function from R to R is an Fσ set
– For monotone functions, the set of discontinuities is at most countable

3. Topological Properties:
– Continuous functions preserve limits of sequences
– The preimage of an open set under a continuous function is open
– The preimage of a closed set under a continuous function is closed
– A continuous function on a closed interval attains all intermediate values (Intermediate Value Theorem)
– Continuous functions preserve connectedness
– Continuous functions preserve compactness
– Continuous functions preserve separability
– The set of continuous functions is path-connected in the uniform topology

4. Analytical Properties:
– A continuous function on a closed, bounded interval is bounded and attains its maximum and minimum
– A continuous function on a closed interval is Riemann integrable
– The composition of continuous functions is continuous
– The sum and product of continuous functions are continuous
– Every continuous function is approximately differentiable almost everywhere
– Continuous functions preserve Borel sets
– A continuous function transforms null sets into null sets if and only if it is absolutely continuous

5. Density Properties:
– Between any two points on the graph of a continuous function, there are infinitely many points
– The graph of a continuous function cannot have “jumps” or “gaps”
– Continuous functions preserve density
– The set of points where a continuous function is differentiable is dense in its domain
– The set of nowhere differentiable continuous functions is residual in C[0,1]

6. Metric Space Properties:
– Continuous functions preserve Cauchy sequences
– For every ε > 0, there exists δ > 0 such that if |x – y| < δ, then |f(x) – f(y)| < ε (ε-δ definition)
– The set of continuous functions forms a complete metric space under the supremum norm
– The space of continuous functions is separable in the compact-open topology
– Continuous functions between metric spaces are uniformly continuous if and only if they preserve totally bounded sets

7. Sequential Properties:
– A function is continuous at a point if and only if it preserves limits of sequences
– If a sequence of continuous functions converges uniformly, its limit is continuous
– Dini’s theorem: if a monotone sequence of continuous functions converges pointwise to a continuous function on a compact space, then the convergence is uniform
– The set of continuous functions is complete under uniform convergence

8. Approximation Properties:
– Any continuous function on a closed interval can be uniformly approximated by polynomials (Weierstrass Approximation Theorem)
– The set of polynomials is dense in the space of continuous functions
– Stone-Weierstrass theorem generalizes polynomial approximation to more general algebras of functions
– Every continuous function can be uniformly approximated by step functions
– Bernstein polynomials provide constructive approximations of continuous functions

9. Fixed Point Properties:
– Any continuous function from [a,b] to [a,b] has at least one fixed point (Brouwer Fixed Point Theorem)
– The set of fixed points of a continuous function is closed
– Schauder fixed point theorem extends to infinite-dimensional spaces
– The fixed point set of a continuous function is compact if the domain is compact

10. Algebraic Structure:
– The space of continuous functions forms a ring under pointwise operations
– It forms a vector space over the real numbers
– The space of continuous functions with the supremum norm forms a Banach space
– It forms a Banach algebra under pointwise multiplication
– The space of continuous functions is a C*-algebra when complex-valued

11. Measure Theory Relationships:
– Every continuous function is Borel measurable
– Continuous functions preserve sets of measure zero if and only if they are absolutely continuous
– The space of continuous functions is dense in Lp spaces for 1 ≤ p < ∞
– Lusin’s theorem relates measurable functions to continuous functions
– Every continuous function is the uniform limit of simple functions

12. Category Theory Properties:
– Continuous functions form a category with topological spaces as objects
– The category of continuous functions preserves products and coproducts
– Continuous functions respect universal properties
– The functor of continuous functions preserves limits and colimits

13. Order-Theoretic Properties:
– Continuous functions preserve order-completeness under certain conditions
– A continuous function on a complete lattice has a fixed point (Knaster-Tarski theorem)
– Continuous functions between ordered topological spaces preserve directed suprema
– The space of continuous functions forms a complete lattice under pointwise ordering

14. Functional Analysis Connections:
– The dual space of continuous functions on a compact space is isomorphic to the space of regular Borel measures
– Continuous functions form a Banach space under various norms
– The spectrum of a continuous function is compact
– The Gelfand transform establishes an isomorphism between commutative C*-algebras and spaces of continuous functions