A Few Wild Functions

Sine Function: {\mathbf{y=\sin x}}

The function {y=\sin x} is beautifully behaved, oscillating regularly along the entire real line {\mathbb{R}} (it is also well-behaved for complex {x} but we won’t consider that here).

The sine function, the essence of good behaviour.

The sine function, the essence of good behaviour.

Chirp Function: {\mathbf{y=\sin x^2}}

Now {y=\sin x^2} is also well-behaved: its oscillations become more rapid as {|x|} increases, but nothing really bad happens. This is called a chirp function, as, if we write it as {y=\sin \omega x} we see that the `frequency’ {\omega=x} increases with {x}. The derivative is {y=2x\cos x^2}, which is regular throughout {\mathbb{R}}.

The function {y=\sin x^2}, a so-called chirp function.

The function {y=\sin x^2}, a so-called chirp function.

Function {\mathbf{y=\sin(1/x)}}

Now let’s get more adventurous and turn the argument of the first function upside-down: {y=\sin(1/x)}. Since {1/x} is not defined for {x=0}, we complete the definition by setting {y(0)=0}. The graph below shows that it is getting wild near the origin. The derivative is {y= (-1/x^2)\cos 1/x}, which is even wilder at {x=0}.

The function {y=\sin 1/x} gets wild near {x=0}.

The function {y=\sin 1/x} gets wild near {x=0}.

Function {\mathbf{y=x\sin(1/x)}}

Let us try to control the wildness at {x=0} by multiplying by {x}, defining {y=x\sin 1/x}. Although {y\longrightarrow0} as {x\longrightarrow0} the derivative is {y^\prime = \sin 1/x - (1/x)\cos 1/x}, which blows up there. Here is the graph of {y=x\sin 1/x}:

The function {y=x\sin 1/x} looks fine near {x=0}, but the derivative blows up.

The function {y=x\sin 1/x} looks fine near {x=0}, but the derivative blows up.

Function {\mathbf{y=x^2\sin(1/x)}}

Now let us further constrain the behaviour of the function near the origin by defining {y=x^2 \sin 1/x}. The derivative is {y^\prime = 2x\sin 1/x - \cos 1/x}, which oscillates wildly as {x\longrightarrow0}, but at least it doesn’t blow up!

The function {y=x^2\sin 1/x} looks fine near {x=0}. The derivative oscillates wildly but does not blows up.

The function {y=x^2\sin 1/x} looks fine near {x=0}. The derivative oscillates wildly but does not blow up.

Function {\mathbf{y=x^2\sin(1/x^2)}}

Finally, we define {y=x^2 \sin 1/x^2} (with {y(0)=0} as usual). This function is bounded on any bounded interval. But {y^\prime = 2x\sin 1/x^2 - (2/x)\cos 1/x^2}, which is unbounded as {x\longrightarrow0}. Let us look at the picture:

The function {y=x^2\sin 1/x^2} looks fine near {x=0}. The derivative is unbounded near {x=0}, yet {y^\prime(0)=0}.

The function {y=x^2\sin 1/x^2} looks fine near {x=0}. The derivative is unbounded near {x=0}, yet {y^\prime(0)=0}.

Non-commuting Limits

The function is squeezed between two parabolas, {y=\pm x^2}; surely, its derivative must vanish at {x=0}. Let us check the fundamental definition

\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{h\rightarrow0} \left[ \frac{y(h)-y(0)}{h} \right] = \lim_{h\rightarrow0} \left[ h \sin 1/h^2 \right] = 0 \,.

Thus the derivative vanishes: {y^\prime(0)=0}. But close to {x=0} it oscillates wildly, and the limit does not exist:

\displaystyle \lim_{x\rightarrow0} y^\prime(x) \ \ \mbox{does not exist} \qquad\mbox{so}\qquad y^\prime(0) \ne \lim_{x\rightarrow0} y^\prime(x) \,.

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RRI-Banner-03Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to RRI.


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