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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