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Magazine Chapter 1: Problem 17
Find the numbers, if any, where the function is discontinuous. \(f(x)=x-[x]\)
Short Answer
Expert verified
The function \(f(x) = x - [x]\) is discontinuous at every integer value, as the left-hand limit (1) and right-hand limit (0) are not equal for integer values of \(x\).
Step by step solution
01
Determine integer values where greatest integer function jumps
Since the greatest integer function jumps at every integer, we will examine the points where \(x\) is an integer. For each of these points, we will look at the left-hand limit and right-hand limit of the function to check for discontinuity.
02
Find the left-hand limit when x is an integer
Let's analyze the left-hand limit for an integer \(n\). We will take the limit as \(x\) approaches \(n\) from the left, denoted by \(\lim_{x \to n^-}\) :
\[\lim_{x \to n^-} (x-[x])\]
Since \(x\) is approaching an integer value \(n\) from the left, \([x]=n-1\) (being the greatest integer less than \(x\)). Hence, the limit becomes:
\[\lim_{x \to n^-} (x-(n-1))\]
As \(x\) approaches \(n\) from the left, the limit becomes:
\[\lim_{x \to n^-} (n - (n - 1)) = 1\]
So, the left-hand limit is 1 when \(x\) approaches an integer \(n\) from the left.
03
Find the right-hand limit when x is an integer
Now, we will analyze the right-hand limit for an integer \(n\). We will take the limit as \(x\) approaches \(n\) from the right, denoted by \(\lim_{x \to n^+}\) :
\[\lim_{x \to n^+} (x-[x])\]
Since \(x\) is approaching an integer value \(n\) from the right, \([x] = n\). Hence, the limit becomes:
\[\lim_{x \to n^+} (x-n)\]
As \(x\) approaches \(n\) from the right, the limit becomes:
\[\lim_{x \to n^+} (n - n) = 0\]
So, the right-hand limit is 0 when \(x\) approaches an integer \(n\) from the right.
04
Compare the left-hand and right-hand limits
We found that the left-hand limit when x is an integer is 1, and the right-hand limit is 0. Since the left-hand limit and the right-hand limit are not equal, the function is discontinuous at every integer value.
05
State the conclusion
The given function \(f(x) = x - [x]\) is discontinuous at every integer value.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
greatest integer function
The greatest integer function, often denoted as \([x]\), is a step function that takes any real number \(x\) and rounds it down to the nearest integer. This means that \([x]\) gives the largest integer less than or equal to \(x\). For example:
- \([3.7] = 3\)
- \([-1.2] = -2\)
- \([5] = 5\)
left-hand limit
In calculus, the concept of a left-hand limit is used to deduce the value that a function approaches as \(x\) approaches a particular point from the left. For the function \(f(x) = x-[x]\), considering the behavior near an integer \(n\), we set our sights on \[ \lim_{x \to n^-} (x-[x]) \]Here, \([x] = n - 1\), because when approaching \(n\) from the left, \(x\) is slightly less than \(n\). Thus, we have\[ \lim_{x \to n^-} (x - (n - 1)) = \lim_{x \to n^-} (x - n + 1) = 1 \]as \(x\) approaches \(n\) from the left.Understanding the left-hand limit is crucial as it tells us what value the function is leaning towards as it gets infinitesimally close to an integer from the left.
right-hand limit
Just like the left-hand limit, a right-hand limit considers what value a function approaches as \(x\) nears a certain point, but this time from the right side. For our function \(f(x) = x-[x]\), and when \(x\) is approaching an integer \(n\) from the right, \[ \lim_{x \to n^+} (x-[x]) \]comes into play. Here, \([x] = n\) since \(x\) is just slightly greater than \(n\). This means we effectively have\[ \lim_{x \to n^+} (x - n) = 0 \]By calculating the right-hand limit, we gain insight into the behavior of the function as we approach an integer from the upper side.
integer values
Integer values play a pivotal role in analyzing the function \(f(x) = x - [x]\) because at these specific points, the greatest integer function jumps, leading to discontinuities. Each point \(x = n\) where \(n\) is an integer is a critical point for examining for breakage in the continuity of the function.For this function:
- Left-hand limit: As you approach an integer \(n\) from the left, \(f(x)\) tends to 1.
- Right-hand limit: As you approach the same integer \(n\) from the right, \(f(x)\) tends to 0.
- Discontinuity at integer \(n\): Because the limits from the left and right at each integer value do not match, the function is discontinuous at every integer value.
