1.1 Simplest possible example 5

• standard manipulations with graphs like shifting, stretching,

etc.:

f(x) + c, f(x + c), cf(x), f(cx).

• composition; for example, the composition of y = |t − 1| and t =

|x| is

f(x) = ||x| − 1|;

✲ ❅

❅

❅

❅

x

0 −1 1

y✻

y = ||x| − 1|

• iterations:

f(x) = ||x| − 1|,

f(f(x)) = |||x| − 1| − 1|,

f(f(f(x))) = ||||x| − 1| − 1| − 1|,

.

.

.

Compare the previous graph of f(x) = ||x|−1| and the one below

of f(f(x)):

✲ ❅ ❅ ❅

❅

x

0 −2 2

y✻

y = |||x| − 1| − 1|

Sketching the 100-th iteration of f becomes an accessible exer-

cise. Can one do the same with polynomials?

And, last but not least, the function y = |x| is not differentiable

(and not analytic!), thus providing a simple, natural, and powerful

example of a non-analytic function. We shall soon see remarkable

implications of this simple observation.