Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\llbracket x-1 \rrbracket\)

Step-by-Step Solution

Answer

The function \(f(x) = \llbracket x-1 \rrbracket\) is continuous for any interval that does not include an integer. The function is not continuous at integer values, as the conditions of continuity are not met (limit from the left is not equal to limit from the right).

Step 1: Identify the Function Type

The floor function \(f(x) = \llbracket x-1 \rrbracket\) is plotted. Observe the graph and note where the function is undefined.

Step 2: Identifying the Intervals of Continuity and Points of Discontinuity

The function is continuous on all intervals except at all integer points, following the expression \(x \in \mathbb{Z}\). At these particular integer points, the limit from the left-hand side is not the same as the limit from the right-hand side, thus marking these points as points of discontinuity.

Step 3: Explain the Discontinuity

The condition of continuity that is not met at these points of discontinuity (integer points) is that, for all \(x \in \mathbb{Z}\), the right-hand limit (from x+) and left-hand limit (from x-) should coincide, which is not the case here.

Key Concepts

Discontinuity in Functions

When we talk about a function being continuous, we mean that you can draw it without lifting your pen. However, when a function suddenly "jumps" or has a break in its graph, it is said to be discontinuous at that point.

For our specific function, which is a floor function, discontinuities occur at every integer point. This is because the left-hand limit and the right-hand limit do not match at these points.

• Left-hand limit: This is the value that the function approaches as you move from left towards the integer point.
• Right-hand limit: This is the value the function approaches as you move from right towards the integer point.

If these two limits are not equal (which they aren't for floor functions), the function is discontinuous at that integer.

Understanding the Floor Function

The function given, \(f(x) = \lfloor x-1 \rfloor\), is known as a floor function. The floor function, denoted by \(\lfloor x \rfloor\), rounds down any real number to the nearest integer less than or equal to it.

This means if you input a real number like 3.7 into the floor function, it outputs 3.

• Behavior: Floor functions show a step-like behavior and are constant between integers.
• Graph: The graph looks like a series of steps, reflecting how it stays the same across a range but drops at integers.

The function \(f(x) = \lfloor x-1 \rfloor\) shifts this behavior by 1 unit, affecting how the function maps inputs to outputs.

Interval Notation

Interval notation is a way of describing the continuous and non-continuous parts of a function by specifying the start and end points of intervals. When using interval notation, pay attention to the symbols used:

• \((a, b)\): Represents all numbers between \(a\) and \(b\), not including \(a\) and \(b\) themselves.
• \([a, b]\): Includes both \(a\) and \(b\) in the interval.

For the floor function, \(f(x) = \lfloor x-1 \rfloor\), it is continuous for all intervals between integers, represented as \((n, n+1)\), where \(n\) is an integer, and discontinuous at the integer points themselves.