REFERENCE SHEET

Limits & continuity

Make approximation statements precise by controlling errors.

Before this Proof techniques

∀ε>0 ∃δ>0: 0<|x−a|<δ ⇒ |f(x)−L|<ε

Definitions

Limit

The displayed condition defines limₓ→ₐ f(x)=L, with x restricted to the domain and a an accumulation point.

Continuity

f is continuous at a if f(a) is defined and limₓ→ₐ f(x)=f(a).

Uniform continuity

For every ε>0 one δ>0 works for every pair of points in the domain whose distance is below δ.

Key theorems

Intermediate value theorem

A continuous real function on [a,b] takes every value between f(a) and f(b).

Extreme value theorem

A continuous real function on a compact interval attains both a maximum and a minimum.

Heine–Cantor

A continuous function on a compact metric space is uniformly continuous.

Notable applications

Root finding

A sign change of a continuous function brackets a root and justifies bisection.

Stable approximation

Uniform continuity controls errors across an entire domain rather than at one input.

Work it out

Try before revealing

01 / Prove

Show limₓ→₂(3x+1)=7 directly.

Hint

Express the output error in terms of |x−2|.

Solution

Given ε>0, choose δ=ε/3. If 0<|x−2|<δ, then |3x+1−7|=3|x−2|<ε.

02 / Find a counterexample

Give a bounded continuous function on (0,1) with no maximum.

Hint

Consider the identity function.