Learn/Algebra/Limits and Continuity

Algebra • Topic 28

Limits and Continuity

While limits are the foundation of calculus, in algebra competitions they appear in functional equations, sequences, and asymptotic behavior analysis.

Definitions

Definition (Limit)

We say

lim_{x \to c} f (x) = L

if for every

ϵ > 0

, there exists a

δ > 0

such that:

0 < ∣ x - c ∣ < δ ⟹ ∣ f (x) - L ∣ < ϵ

Definition (Continuity)

A function

f

is continuous at

c

if:

$f (c)$ is defined.
$lim_{x \to c} f (x)$ exists.
$lim_{x \to c} f (x) = f (c)$ .

Olympiad Applications

1. Solving Functional Equations

Continuity is a powerful constraint.

Cauchy's Equation: $f (x + y) = f (x) + f (y)$ .

Without continuity: Wild solutions exist (using Hamel bases).

With continuity: Only $f (x) = c x$ .

2. Intermediate Value Theorem (IVT)

f

is continuous on

[a, b]

and

k

is between

f (a)

and

f (b)

, then there exists

c \in [a, b]

such that

f (c) = k

Use case: Proving a polynomial has a real root.

3. Extreme Value Theorem

f

is continuous on a closed interval

[a, b]

, it attains a maximum and minimum.

Use case: Justifying that an extremum exists before using inequalities to find it.

Limits of Sequences

For a sequence

a_{n}

lim_{n \to \infty} a_{n} = L

means terms get arbitrarily close to

L

Monotone Convergence: A bounded, monotonic sequence always converges.

Practice Problems

Exercise (Problem 1)

Let

f : R \to R

be continuous such that

f (x) = f (x /2)

for all

x

. Prove

f

is constant.

Exercise (Problem 2)

Evaluate

lim_{n \to \infty} (1 + \frac{x}{n})^{n}

Exercise (Problem 3)

Let

f (x) = x^{3} - 3 x + 1

. Prove that

f (x)

has 3 distinct real roots.