Functions: Continuity of a Function

Understand the concept of continuity, explore continuous and discontinuous functions, and practice finding points of discontinuity.

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Learning Guide: The Coordinate System

Learning Guide: Continuity of a Function

A function

f(x)

is said to be continuous if its graph is a single connected curve with no breaks, holes, or jumps. Intuitively, you can think of a continuous function as one whose graph can be drawn without lifting your pencil.

Continuity of a Function

A function

f(x)

is continuous at a point

x = a

if:
1.

f(a)

is defined.
2. The graph approaches the same

y

-value from both the left and the right.
3. The function value at

a

matches this

y

-value (no jumps or holes).

Continuous and Discontinuous Functions

1. Continuous Functions

A function is continuous if its graph has no breaks, holes, or jumps. Intuitively, you can think of its graph as one that can be drawn without lifting your pencil.

Polynomial

Polynomials: Functions like $f(x) = x^2 - 3$ or $f(x) = 2x + 1$ are continuous everywhere.

Absolute Value

Absolute Value: $f(x) = |x - 2|$ is continuous everywhere.

2. Discontinuous Functions

A function is discontinuous if its graph has breaks, jumps, holes, or shoots off to infinity.

2.1 Jump Discontinuity

The function jumps from one height to another at a specific point. The left-hand limit and right-hand limit exist but are not equal.

Jump Discontinuity

Income Tax Brackets: An everyday example is tax rates, where the tax rate jumps (e.g., from 10% to 20%) once income crosses a certain threshold.

2.2 Hole Discontinuity (Removable)

The function is defined and continuous everywhere except at a single point, where there is a missing point (hole).

Hole Discontinuity

Rational Holes: $f(x) = \frac{x^2-4}{x-2}$ is undefined and has a hole at $x = 2$ , though it looks like the line $y = x + 2$ everywhere else.

2.3 Infinite Discontinuity (Vertical Asymptote)

The function shoots up or down towards infinity as it approaches a certain value, creating a vertical split in the graph.

Infinite Discontinuity

Unbounded Split: In $f(x) = \frac{1}{x-3}$ , as $x$ gets closer to 3, division by a tiny number makes the function value shoot up to $+\infty$ or down to $-\infty$ (where the symbol $\infty$ means infinity — growing larger and larger without limit). In mathematics, this vertical boundary line that the graph gets infinitely close to, but never touches, is called a vertical asymptote.
Because of this split, the function is discontinuous at $x = 3$ .

Learning Topics

Frequently Asked Questions

Why does the x-coordinate always come first in an ordered pair?

By mathematical convention, coordinates are always written in alphabetical order as

(x, y)

. This standardized order ensures that anyone around the world can communicate and locate points on a coordinate plane consistently without ambiguity.

What does it mean for a function to be continuous?

Intuitively, it means you can draw the function's graph without lifting your pencil. Formally, a function must be defined at the point, and the graph must connect smoothly without any gaps, jumps, or holes.

How do you find points where a function is not continuous?

Look for inputs that make the function undefined (like division by zero). For example,

f(x) = \frac{x^2 - 16}{x - 4}

is discontinuous at

x = 4

because you cannot divide by zero, creating a hole in the graph.