Continuity of Functions

Continuity means you can draw the whole graph in one stroke without lifting your pen off the paper. It is a smooth curve with no sudden breaks, no staircase jumps, and no punched-out holes anywhere. Strictly speaking, a function is continuous at a point only when its limit and its actual value match exactly there. Tap the continuous, jump, and removable buttons to compare how and where an otherwise smooth graph can fall apart.

Intuition of Continuity

✏ What is continuity?

①"A graph you can draw without lifting the pen"

②No breaks, no jumps, no holes — a smooth curve

③Mathematically: three conditions must hold simultaneously

Continuous vs Discontinuous

Three Conditions

continuous at x = a ⟺ ①f(a) exists ②lim_x→a f(x) exists ③lim_x→a f(x) = f(a)

If any condition fails ⇒ discontinuous

🔍 Types of Discontinuity

①Jump: left limit ≠ right limit (step function)

②Removable: limit exists but f(a) differs from it (hole)

③Infinite: the limit itself is ±∞ (vertical asymptote)

Properties of Continuous Functions

Arithmetic

If f, g continuous, then f ± g, f · g, f/g (g ≠ 0) are continuous

Sums, products, quotients of continuous functions stay continuous

Composition

f continuous at a, g continuous at f(a) ⇒ g ∘ f continuous at a

Composition of continuous functions is continuous

Intermediate Value Theorem

Target k0.5

IVT

f(a) < k < f(b) ⇒ ∃c ∈ (a, b), f(c) = k

A continuous function must hit every intermediate value — root-existence tool

💡 Using IVT

①"If f(a) < 0 and f(b) > 0, there is c ∈ (a, b) with f(c) = 0"

②Essential for proving existence of real roots

③f must be continuous on [a, b]

Wrap-up

Definition

lim_x→a f(x) = f(a)

Limit equals function value ⇒ continuous

🎯 Exam Points

①Three-condition continuity: f(a) exists + limit exists + they're equal

②Polynomials, trig, exp/log are continuous on their domains

③Discontinuity check: compare left and right limits

④IVT: existence of roots — "sign change ⇒ root"

⑤Unknown-coefficient problems: use continuity (left=right=f(a)) to fix constants

Worked Examples & Past Exam

Example 1

For f(x) = (x² − 1)/(x − 1) (x ≠ 1) and f(1) = k, find the constant k that makes f continuous at x = 1.

For continuity at x=1, we need lim_x→1 f(x) = f(1) = k.

x² - 1x - 1 = x + 1 (x ≠ 1)

Find the limit and set it equal to k.

lim_x→1 (x + 1) = 2 = k

▸ k = 2

A removable discontinuity becomes continuous when the value equals the limit.

Example 2

For f(x) = x² − 3, explain that the equation f(x) = 0 has at least one real root in the interval (1, 2).

f is a polynomial, so it is continuous on [1, 2]; check the signs at the endpoints.

f(1) = -2 < 0, f(2) = 1 > 0

The signs differ, so by the IVT there is c in (1, 2) with f(c) = 0.

f(1) < 0 < f(2) ⟹ f(c) = 0, c ∈ (1, 2)

▸ A real root exists in (1, 2).

If a continuous function changes sign across an interval, a root lies between (IVT).

2022 KICE mock exam Math type, adapted

For f(x) = { 2x + a (x ≤ 1), x² + 2 (x > 1) }, find the constant a if f is continuous at x = 1.

①1

②3

③2

④-1

⑤0

▸ ① 1

For continuity at x=1, left limit = right limit = f(1).

left = 2(1) + a = 2 + a, right = 1² + 2 = 3

Set them equal.

2 + a = 3 ⟹ a = 1

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Limit of a Function

Derivative & Differentiation

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