Limit and Continuity in Metric Spaces

Study Notes

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)).

Description

This quiz covers the definitions and properties of limits and continuity in metric spaces. It includes discussions on limit notation, uniqueness of limits, and continuity concepts for functions defined in Euclidean spaces. Test your understanding of these foundational concepts in real and complex analysis.