math 104: limits of functions and continuity

continuity at a point

f: E → Y is continuous at p ∈ E if for every ε > 0 there is a δ > 0 such that d_Y(f(x), f(p)) < ε for all x ∈ E with d_X(x, p) < δ

at an isolated point, continuity is automatic

continuous on E

continuous at every point of E

compositions of continuous functions

are continuous

sums, products, quotients (nonzero denominator) of continuous functions

are continuous

a function into ℝᵏ is continuous

if and only if each coordinate function is

polynomials, rational functions (where defined), and |x|

are continuous

global criterion / topological characterization

f: X → Y is continuous if and only if f⁻¹(V) is open in X for every open V ⊆ Y; equivalently, preimages of closed sets are closed (this is about preimages; images of open sets need not be open)

the continuous image of a compact set is compact

image = output set f(K) that you get when you apply a continuous function to a compact set K

extreme value theorem

a continuous real function on a compact metric space is bounded and attains its supremum/maximum and infimum/minimum

a continuous bijection of a compact metric space onto a metric space has continuous inverse

compactness of the domain is essential

uniform continuity

f: X → Y is uniformly continuous if for every ε > 0 there is a δ > 0 such that d_Y(f(p), f(q)) < ε for all p, q ∈ X with d_X(p, q) < δ. One δ works for all locations, whereas in ordinary continuity δ may depend on the point

a continuous function on a compact set is uniformly continuous

requires compactness

image

the set of all output values that a function produces when you plug in every possible input from its domain

intermediate value theorem

If f is continuous on [a, b] and f(a) < c < f(b), then f(x) = c for some x ∈ (a, b)

one-sided limits

f(x+) = q if f(tₙ) → q for every sequence tₙ > x with tₙ → x. f(x−) is defined using tₙ < x. At an interior point, lim_{t→x} f(t) exists if and only if f(x+) = f(x−)

simple discontinuity

first kind: both one-sided limits exist but f is discontinuous at x; otherwise the discontinuity is of the second kind

monotone functions and one-sided limits

a monotone function has one-sided limits everywhere, hence only simple discontinuities, and the set of its discontinuities is at most countable