Use Interval Notation to Indicate Where Fx Is Continuous Calculator

Interval notation is a concise way to represent sets of real numbers. When analyzing functions, it's essential to determine where a function is continuous. This calculator helps you express the continuity of a function using interval notation.

What is Interval Notation?

Interval notation is a method of representing a set of real numbers using parentheses and brackets. The most common types of intervals are:

(a, b): All numbers between a and b, not including a and b
[a, b]: All numbers between a and b, including a and b
(a, b]: All numbers between a and b, not including a but including b
[a, b): All numbers between a and b, including a but not including b
(a, ∞): All numbers greater than a
(-∞, b): All numbers less than b
(-∞, ∞): All real numbers

Interval notation is particularly useful when describing the domain and continuity of functions.

How to Determine Where a Function is Continuous

A function is continuous at a point if three conditions are met:

The function is defined at that point
The limit of the function exists at that point
The limit equals the function's value at that point

To determine where a function is continuous, you need to:

Identify any points where the function is undefined
Find any points where the limit does not exist
Check for points where the limit exists but doesn't equal the function value

Once you've identified these points, you can express the continuity of the function using interval notation.

Common Continuity Intervals

Many standard functions have well-known continuity intervals:

Polynomial functions: Continuous everywhere (-∞, ∞)
Rational functions: Continuous everywhere except where the denominator is zero
Exponential functions: Continuous everywhere (-∞, ∞)
Trigonometric functions: Continuous everywhere (-∞, ∞)
Absolute value functions: Continuous everywhere (-∞, ∞)

For more complex functions, you'll need to analyze them individually to determine their continuity intervals.

Example Problems

Example 1: Rational Function

Consider the function f(x) = (x² - 4)/(x - 2).

First, simplify the function: f(x) = (x + 2)(x - 2)/(x - 2) = x + 2 for x ≠ 2.

The function is undefined at x = 2. Therefore, it's continuous on (-∞, 2) ∪ (2, ∞).

Example 2: Piecewise Function

Consider the function:

f(x) = { x² + 1, if x ≤ 0 x + 3, if x > 0 }

Analyze each piece:

For x ≤ 0: x² + 1 is a polynomial, continuous everywhere
For x > 0: x + 3 is a polynomial, continuous everywhere
At x = 0: Both pieces are defined and equal (f(0) = 1)

Therefore, the function is continuous on (-∞, ∞).

FAQ

What is the difference between open and closed intervals?: An open interval does not include its endpoints (parentheses), while a closed interval includes its endpoints (brackets).
How do I know if a function is continuous at a point?: A function is continuous at a point if it's defined there, the limit exists, and the limit equals the function value.
Can a function be continuous on an open interval but not closed?: Yes, a function can be continuous on an open interval but not at the endpoints. For example, f(x) = 1/x is continuous on (0, ∞) but not at x = 0.
What's the difference between continuity and differentiability?: A function can be continuous but not differentiable (like |x| at x = 0), or differentiable but not continuous (which is impossible).
How do I express continuity over multiple intervals?: Use the union symbol (∪) to combine intervals. For example, (-∞, 2) ∪ (2, ∞) means all real numbers except 2.