Limits 3

What does it mean for f:A⊆ℝ→ℝ to be continuous at an accumulation point x₀∈A?

f is continuous at x₀ if limₓ→ₓ₀ f(x)=f(x₀).

How is continuity defined at an isolated point of the domain?

By convention, f is continuous at every isolated point of its domain. The ε-δ definition also holds automatically there.

What does it mean for f to be continuous on its domain A?

f is continuous on A if it is continuous at every point x₀∈A.

What three conditions must hold for f to be continuous at x₀?

f(x₀) must be defined; the limit of f(x) as x→x₀ must exist; and that limit must equal f(x₀).

How is continuity defined at a boundary point of the domain?

The limit is taken using only points in the domain. At an interval endpoint, this is a one-sided limit.

What is the ε-δ definition of continuity at x₀∈A?

For every ε>0, there exists δ>0 such that for every x∈A, if |x−x₀|<δ, then |f(x)−f(x₀)|<ε.

What is the sequential characterization of continuity at x₀?

f is continuous at x₀ if and only if, for every sequence (xₙ) in A with xₙ→x₀, we have f(xₙ)→f(x₀).

Where are elementary functions continuous?

Each elementary function is continuous on its natural domain. For example: polynomials on ℝ; rational functions where the denominator is nonzero; exponentials on ℝ; logarithms on (0,+∞); and sin and cos on ℝ.

Where should continuity be checked for a piecewise function whose branches are continuous on their own domains?

Check the breakpoints between the branches; these are the additional points where continuity may fail.

What is a discontinuity of f at x₀?

f is discontinuous at x₀ if it is not continuous there.

What is a removable (hole) discontinuity at x₀?

The finite limit limₓ→ₓ₀ f(x) exists, but it is not equal to f(x₀).

What is a jump discontinuity at x₀?

The left-hand and right-hand limits both exist and are finite, but they are unequal.

What is an essential (non-removable) discontinuity at x₀?

At least one one-sided limit is infinite or does not exist.

What is a non-removable discontinuity?

A discontinuity that cannot be removed by changing only the function’s value at the point; jump and essential discontinuities are non-removable.

What does the algebra theorem for continuous functions state?

If f and g are continuous at x₀, then αf, f+g, and fg are continuous at x₀ for every constant α. Also, f/g is continuous at x₀ if g(x₀)≠0.

What does the composition theorem for continuous functions state?

If f is continuous at x₀ and g is continuous at f(x₀), then g∘f is continuous at x₀, provided the composition is defined.

What does Bolzano’s zero theorem state?

If f:[a,b]→ℝ is continuous and f(a)f(b)≤0, then there is some c∈[a,b] such that f(c)=0.

When is the zero guaranteed by Bolzano’s theorem unique?

If f is strictly monotonic on [a,b], the zero is unique.

What does Bolzano’s theorem imply about intersections of two continuous functions?

If f,g:[a,b]→ℝ are continuous, f(a)≥g(a), and f(b)≤g(b), then there is some c∈[a,b] such that f(c)=g(c).

When is the intersection guaranteed by the two-function version of Bolzano’s theorem unique?

If f is strictly decreasing and g is strictly increasing, the intersection point is unique.

What is a market equilibrium in terms of demand and supply?

A price-quantity pair (p̂,q̂) is a market equilibrium if q̂=D(p̂)=S(p̂).

What does the market-equilibrium existence theorem state?

If continuous demand D and supply S on [a,b] satisfy D(a)≥S(a) and D(b)≤S(b), then a market equilibrium exists.

When is the market equilibrium guaranteed to be unique?

If demand is strictly decreasing and supply is strictly increasing, the equilibrium is unique.

What does Weierstrass’s extreme value theorem state?

A continuous function on a compact set A⊆ℝ attains both a maximum and a minimum: there are points in A where its largest and smallest values are reached.

What does “f attains its maximum and minimum” mean?

There are points x₁,x₂ in the domain such that f(x₁)=maxₓ∈A f(x) and f(x₂)=minₓ∈A f(x).

What does Darboux’s intermediate value theorem state?

If f:[a,b]→ℝ is continuous, and m=minₓ∈[a,b] f(x), M=maxₓ∈[a,b] f(x), then every z∈[m,M] equals f(c) for some c∈[a,b].

When is the point c in Darboux’s theorem guaranteed to be unique?

If f is strictly monotonic, the point c satisfying f(c)=z is unique.

What is the relationship between continuity, injectivity, and strict monotonicity on an interval?

A continuous function f:I→ℝ on an interval I is injective if and only if it is strictly monotonic.