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Magazine Chapter 2: Problem 13
If \(f(x)=x+\\{-x\\}+[x]\), where \([x]\) and \(\\{x\\}\) denotes greatest integer function and fractional part function respectively, then find the number of points at which \(f(x)\) is both discontinuous and non-differentiable in \([-2,2]\)
Short Answer
Expert verified
In the interval \([-2,2]\), there is only 1 point where \(f(x)\) is discontinuous and non-differentiable, which is at \(x=1\).
Step by step solution
01
Identify intervals of integer and fractional parts
We can break down the interval \([-2,2]\) into sub-intervals based on integer values:
1. \([-2, -1)\)
2. \([-1, 0)\)
3. \([0, 1)\)
4. \([1, 2]\)
For each sub-interval, we will separately analyze the function for discontinuity and non-differentiability.
02
Analyze the interval \([-2, -1)\)
In \([-2, -1)\), we have \([x] = -2\) and \(\{-x\} = x + 2\).
So, \(f(x) = x + (x + 2) - 2 = 2x\).
The function \(f(x) = 2x\) is continuous and differentiable on \([-2, -1)\). The potential discontinuity could only appear at \(x=-1\).
03
Analyze the interval \([-1, 0)\)
In \([-1, 0)\), we have \([x] = -1\) and \(\{-x\} = x + 1\).
So, \(f(x) = x + (x + 1) - 1 = 2x\).
The function \(f(x) = 2x\) is continuous and differentiable on \([-1, 0)\). The potential discontinuity could only appear at \(x=0\).
04
Analyze the interval \([0, 1)\)
In \([0, 1)\), we have \([x] = 0\) and \(\{-x\} = -x\).
So, \(f(x) = x - x = 0\).
The function \(f(x) = 0\) is continuous and differentiable on \([0, 1)\). The potential discontinuity could only appear at \(x=1\).
05
Analyze the interval \([1, 2]\)
In \([1, 2]\), we have \([x] = 1\) and \(\{-x\} = -x + 1\).
So, \(f(x) = x - x + 1 = 1\).
The function \(f(x) = 1\) is continuous and differentiable on \([1, 2]\). There is no potential discontinuity in this interval.
Now we have analyzed all sub-intervals. The potential discontinuities and non-differentiability points lie at \(x = -1, 0, 1\). We need to check the continuity and differentiability at these points.
06
Analyze discontinuity and non-differentiability points
At \(x=-1\), we get \(f(-1^-) = 2(-1) = -2\) and \(f(-1^+) = 2(-1) = -2\). Since the left limit and the right limit are equal, the function is continuous at \(x=-1\).
At \(x=0\), we get \(f(0^-) = 2(0) = 0\) and \(f(0^+) = 0\). Since the left limit and the right limit are equal, the function is continuous at \(x=0\).
At \(x=1\), we get \(f(1^-) = 0\) and \(f(1^+) = 1\). Since the left limit and the right limit are not equal, the function is discontinuous at \(x=1\). Note that the function is also non-differentiable at this point.
07
Counting the points where the function is discontinuous and non-differentiable
In the interval \([-2,2]\), we found that the function \(f(x)\) is discontinuous and non-differentiable only at \(x=1\). So, there is only 1 point where \(f(x)\) is discontinuous and non-differentiable.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Discontinuous Functions
A function is termed discontinuous if there is an abrupt change in the function's value, breaking its graph. For a function to be continuous at a point, the limit of the function as it approaches that point must equal its value at the point. Discontinuity in a function occurs when:
- The left-hand limit and right-hand limit at a point do not match.
- The limit exists but is different from the value of the function at that point.
Non-differentiable Points
A point on a function is non-differentiable if the derivative does not exist at that point. This could happen if the function has a sharp corner, cusp, vertical tangent, or is discontinuous. Differentiability requires the function to be smooth enough at a point so that the slope of the tangent line is well-defined. If a function is continuous at a point, it might still be non-differentiable due to a sharp turn. In our function \(f(x)\), after examining possible non-differentiable points, the point \(x = 1\) was found to be both non-differentiable and discontinuous, contributing to function irregularity.
Greatest Integer Function
The greatest integer function, denoted as \([x]\), returns the largest integer less than or equal to \(x\). For example, \([3.7] = 3\) and \([-1.2] = -2\). This function inherently has discontinuity at integer points since the value jumps significantly from one integer to the next. Therefore, in the range \([-2,2]\), this function contributed to sharpening our attention toward points like \(x = -1, 0, 1\), checking for significant changes or jumps that characterize discontinuities.
Fractional Part Function
The fractional part function, denoted as \(\{x\}\), is defined as \(x - [x]\). This function gives the decimal or fractional part of a number. For instance, if \(x = 3.7\), then \(\{x\} = 0.7\) and \(x = -1.2\), then \(\{x\} = 0.8\). This function is continuous across its domain because it outputs values between 0 and just under 1. The role of \(\{x\}\) in the given function \(f(x)\) ensured continuity within intervals but contributed to its unique value changes at potential points, supporting the analysis of discontinuities and differentiability at integer transitions.
