Discussion
Consider the function
\[ f(x) = \begin{cases} x \sin\left(\dfrac{1}{x}\right), & \text{if } 0 \lt x \le 1 \\ 0, & \text{if } x = 0 \end{cases} \]
Then which of the following statements is true ?
Answer: A
The true statement is \(f\) is Riemann integrable.
Since \(|x \sin(1/x)| \le |x| \to 0\), \(f\) is continuous at 0, and it is clearly continuous on (0, 1]. A continuous function on [0, 1] is bounded and Riemann integrable.
The other statements fail. \(f\) is continuous at 0, as just shown. It is not differentiable at 0, because \((f(h) - f(0))/h = \sin(1/h)\) has no limit. It is not of bounded variation: between consecutive extrema at \(x = 2/((2k+1)\pi)\) the function swings by about \(4/((2k+1)\pi)\), and these swings form a divergent harmonic-type series.
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