**Sine Function:**

The function is beautifully behaved, oscillating regularly along the entire real line (it is also well-behaved for complex but we won’t consider that here).

**Chirp Function:**

Now is also well-behaved: its oscillations become more rapid as increases, but nothing really bad happens. This is called a chirp function, as, if we write it as we see that the `frequency’ increases with . The derivative is , which is regular throughout .

**Function**

Now let’s get more adventurous and turn the argument of the first function upside-down: . Since is not defined for , we complete the definition by setting . The graph below shows that it is getting wild near the origin. The derivative is , which is even wilder at .

**Function**

Let us try to control the wildness at by multiplying by , defining . Although as the derivative is , which blows up there. Here is the graph of :

**Function**

Now let us further constrain the behaviour of the function near the origin by defining . The derivative is , which oscillates wildly as , but at least it doesn’t blow up!

**Function**

Finally, we define (with as usual). This function is bounded on any bounded interval. But , which is unbounded as . Let us look at the picture:

**Non-commuting Limits**

The function is squeezed between two parabolas, ; surely, its derivative must vanish at . Let us check the fundamental definition

Thus the derivative vanishes: . But close to it oscillates wildly, and the limit does not exist:

* * * * * *

Peter 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.