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.
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