Limit and Continuity in Metric Spaces

Limit and Continuity Definitions

• In metric spaces X and Y, if E ⊂ X, and p is a limit point of E, f: E → Y, limit notation is defined as f(x) → q as x → p when for every ε > 0, a δ > 0 exists such that dY(f(x), q) < ε for x ∈ E where 0 < dX(x, p) < δ.
• If X or Y is a Euclidean space Rk or the complex plane, the metrics dX and dY are represented by absolute values or norms of differences.
• The limit condition lim x→p f(x) = q is equivalent to saying that for every sequence {pn} in E (pn ≠ p, lim n→∞ pn = p), lim n→∞ f(pn) = q.

Properties of Limits and Functions

• The limit at point p is unique if it exists.
• For complex functions f and g defined on E, if lim x→p f(x) = A and lim x→p g(x) = B, then:
  • lim x→p (f+g)(x) = A + B
  • lim x→p (fg)(x) = AB
  • lim x→p (f/g)(x) = A/B if B ≠ 0.

Continuity Concepts

• A function f is continuous at p if for every ε > 0, there exists a δ > 0 such that dY(f(x), f(p)) < ε for all x ∈ E with dX(x, p) < δ.
• A function is continuous on E if it is continuous at every point in E.
• If p is an isolated point of E, any function with E in its domain is continuous at p.

Conditions for Continuity

• For f to be continuous at p, it must be defined at p.
• f is continuous at a limit point p if and only if lim x→p f(x) = f(p).
• The composition of continuous functions is also continuous.

Function Mappings and Topology

• A mapping f: X → Y is continuous if f^(-1)(V) is open for every open set V in Y.
• It is continuous if f^(-1)(C) is closed for every closed set C in Y.
• For continuous complex functions f and g on a metric space X, both f + g and f * g are continuous.

Boundedness and Compactness

• A mapping f from E into Rk is bounded if |f(x)| ≤ M for some real number M across all x in E.
• Continuous mappings from compact metric spaces to metric spaces yield compact images.
• If f is continuous on a compact set X, it is both closed and bounded.

Extreme Value Theorem

• For a continuous real function on a compact metric space, maximum and minimum are attained at points p and q in the space, respectively.

Inverse Mappings

• A continuous 1-1 mapping of a compact metric space X onto Y implies that the inverse mapping is also continuous from Y back to X.

Uniform Continuity

• A mapping f is uniformly continuous if the condition dY(f(p), f(q)) < ε holds for all p, q in X such that dX(p, q) < δ.
• Every uniformly continuous function is inherently continuous.
• Continuous functions mapping compact metric spaces to metric spaces are uniformly continuous.

Noncompactness Properties

• Noncompact sets in R¹ can possess continuous functions that are unbounded or continuous and bounded without achieving maxima.
• For bounded noncompact sets, continuous functions may exist that are not uniformly continuous.

Connectivity of Images

• If E is a connected subset of X and f is continuous, then the image f(E) remains connected.

Intermediate Value Theorem

• A continuous real function f on an interval [a, b] such that f(a) < f(b) ensures the existence of at least one point x in (a, b) where f(x) = c for any c in the interval (f(a), f(b)).