Question

Show that the function \(f(x)=\left\\{\begin{array}{cl}\frac{x \ell n x}{1-x}, & 0

Short answer

Answer: Yes, the function \(f(x)\) is Riemann integrable on the interval \([0,1]\).

Step-by-step solution

Check continuity

First, let's check if the function is continuous at its junction points. When \(0<x<1\) we have \(f(x)=\frac{x\ln x}{1-x}\), when \(x=0\) we have \(f(x)=0\), when \(x=1\) we have \(f(x)=-1\). To check the continuity at \(x=0\), find the limit as \(x\) approaches \(0\): \(\lim_{x\to 0^+}\frac{x\ln x}{1-x}\). Using L'Hopital's rule, differentiate the numerator and the denominator with respect to \(x\): \(\lim_{x\to 0^+}\frac{\ln x + 1}{1} = 0.\) Since the limit value is equal to \(f(0)\), the function is continuous at \(x=0\). Now, let's check the continuity at \(x=1\). Find the limit as \(x\) approaches \(1\): \(\lim_{x\to 1^-}\frac{x\ln x}{1-x}\). Using L'Hopital's rule again, differentiate the numerator and the denominator with respect to \(x\): \(\lim_{x\to 1^-}\frac{\ln x + 1}{1}=0 \neq -1.\) Since the limit value is not equal to \(f(1)\), the function is not continuous at \(x=1\). However, this does not necessarily mean that the function is not integrable.

Check for Riemann integrability

Now, let's check if the function is Riemann integrable. A function is Riemann integrable if its set of discontinuities has a measure of zero. Since the only point of discontinuity for this function is at \(x=1\), the set of its discontinuities has a measure of zero. This means that the function is Riemann integrable, even though it is not continuous at \(x=1\).

Conclusion

We showed that the function \(f(x)\) is continuous at \(x=0\) and discontinuous at \(x=1\). However, since its set of discontinuities has a measure of zero, meaning the function is Riemann integrable on the interval \([0,1]\).

Key concepts

Continuity in Calculus

Continuity is a fundamental concept in calculus that describes a property of functions. A function is said to be continuous at a point if the following three conditions are met: first, the function is defined at the point; second, the limit of the function as it approaches the point exists; and third, the limit and the function value at that point are equal.

For example, when evaluating whether a function like \(f(x)=\frac{x \ln x}{1-x}\) is continuous at a point such as \(x=0\), we look at the behavior of the function as it gets arbitrarily close to that point. If we can show that the limit as \(x\) approaches 0 from the right (denoted as \(x \to 0^+\)) is equal to the function's value at 0, which in this case is 0, then the function is continuous there. This continuity at the boundaries of an interval is essential when determining the Riemann integrability of a function.

L'Hopital's Rule

L'Hopital's rule is an incredibly useful tool for finding limits that present an indeterminate form such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). This rule states that if the limits of the numerator and the denominator of a function both approach 0 or \(\pm\infty\), the limit of the function can be found by differentiating the numerator and the denominator separately and then taking the limit of their quotient.

For instance, if we encounter a limit like \(\lim_{x\to 0^+}\frac{x\ln x}{1-x}\), L'Hopital's rule authorizes us to differentiate the top (\(\ln x + 1\)) and the bottom (constantly 1) and examine the limit again. Sometimes, you may need to apply L'Hopital's rule multiple times or simplify the function further to get a clear answer. This rule is indispensable in calculus, especially when dealing with complex functions and is a critical step in verifying the conditions for Riemann integrability.

Limit of a Function

The concept of limits is at the heart of calculus. It describes the behavior of a function as the input (or the 'x' value) approaches a certain value. A limit does not concern itself directly with the function value at that point, but rather what value the function is approaching.

For the function \(f(x)\), the limit as \(x\) approaches some value 'a' is symbolically represented as \(\lim_{x \to a}f(x)\). Limits can be one-sided, meaning we approach 'a' from only one side – either from the left (\(x \to a^-\)) or the right (\(x \to a^+\)). Understanding limits is crucial, as they are used to define both derivatives (the rate of change of functions) and integrals (the accumulation of quantities). In the context of Riemann integrals, limits help determine the continuity and integrability of functions.

Riemann Integral

The Riemann integral is a method to calculate the area underneath a curve on a graph, essentially finding the total accumulated value represented by the function over a certain interval. A function is Riemann integrable if you can approximate its area with a series of rectangles (called Riemann sums) and these approximations become more accurate as the widths of the rectangles get infinitesimally small.

A key requirement for a function to be Riemann integrable over an interval is that it should not have too many discontinuities. Precisely, the set of its discontinuity points should have a measure of zero. This condition is satisfied even if a function has isolated discontinuities; it can still be integrated. In the case of our exercise, the discontinuity at \(x=1\) is not a barrier to Riemann integrability, demonstrating that continuity everywhere is not a necessity for integrability over an interval.