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!

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

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, ; 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.