Graph of the Function $y = x\sin\dfrac{1}{x}$

$y = x\sin\dfrac{1}{x}$ multiplies $\sin\dfrac{1}{x}$ by $x$. That factor damps the amplitude, so the wild oscillation near the origin is crushed down to zero along with its height. It is the squeeze theorem made into a shape, and also the standard example showing that continuity does not guarantee differentiability.

Domain and symmetry

The domain is $x \neq 0$. The function turns out to be even.

The graph is therefore symmetric about the $y$-axis, exactly as the rule that a product of two odd functions is even predicts.

The squeeze

From $\left|\sin\dfrac{1}{x}\right| \leq 1$ the following bound holds¹.

The graph is thus confined inside the wedge formed by the lines $y = x$ and $y = -x$. Pressed toward $0$ from both sides, the limit as $x \to 0$ is $0$. The oscillation itself never stops and its frequency never drops; only its amplitude vanishes. The curve touches the sides of the wedge at $x = \dfrac{2}{(2n+1)\pi}$, where $\left|\sin\dfrac{1}{x}\right| = 1$.

Continuous but not differentiable

Defining $f(0) = 0$ makes the function continuous at the origin, since that is the limit. It is nevertheless not differentiable there, because the difference quotient behaves as follows.

This has no limit as $h \to 0$. One line of computation separates continuity from differentiability.

Raising the power of the factor $x$ tames the behavior at the origin one step at a time.

Behavior far out

Setting $u = \dfrac{1}{x}$ shows that the function is nothing other than the sinc function composed with a reciprocal.

Since $x \to \pm\infty$ corresponds to $u \to 0$, we get $y \to 1$ and the line $y = 1$ is a horizontal asymptote. All the infinitely many oscillations that sinc spreads across $u > 0$ are compressed into the immediate neighborhood of the origin.

Range and extrema

For $u \neq 0$ we always have $\dfrac{\sin u}{u} < 1$, and its first minimum occurs at the solution of $\tan u = u$ near $u \approx 4.4934$, where the value is about $-0.2172$. Transferring this back, the minimum of our function is about $-0.2172$, attained at $x = \dfrac{1}{4.4934} \approx 0.2225$, and the range is $-0.2172 \leq y < 1$. The value $1$ is never reached, precisely because $\dfrac{\sin u}{u} < 1$ always holds.

Zeros

The zeros are at $x = \dfrac{1}{n\pi}$, where $\sin\dfrac{1}{x} = 0$, exactly the same as for $\sin\dfrac{1}{x}$. Multiplying by $x$ moves none of them, and they still accumulate at the origin.

Significance

In a first analysis course this function is the standard first application of the squeeze theorem. It is at the same time a concrete curve that is continuous at a point yet has no tangent there, making visible that differentiability is a stronger condition than continuity.

1. Squeeze theorem, Wikipedia