A Smooth Function Which Is Nowhere Real Analytic
A more pathological example, of an infinitely differentiable function which is not analytic at any point can be constructed by means of a Fourier series as follows. Let A:={2n : n ∈ N } be the set of all powers of 2, and define for all x ∈ R
Since the series converge for all n ∈ N, this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test, and of the theorem of limit under the sign of derivative. Moreover, for any dyadic rational multiple of π, that is x:=π p/q with p ∈ N and q ∈ A, and for all order of derivation n ∈ A, n ≥ 4 and n > q we have
where we used the fact that cos(kx)=1 for all k > q. As a consequence, at any such x ∈ R
so that the radius of convergence of the Taylor series of f at x is 0 by the Cauchy-Hadamard formula . Since the set of analyticity of a function is an open set, and since dyadic rationals are dense, we conclude that f is nowhere analytic in R.
Read more about this topic: Non-analytic Smooth Function
Famous quotes containing the words smooth, function, real and/or analytic:
“or the warm soft side
Of the resigning yet resisting bride.
The kiss of virgins first-fruits of the bed;
Soft speech, smooth touch, the lips, the maidenhead;
These and a thousand sweets could never be
So near or dear as thou wast once to me.”
—Robert Herrick (1591–1674)
“The function of muscle is to pull and not to push, except in the case of the genitals and the tongue.”
—Leonardo Da Vinci (1425–1519)
“Fine art is the subtlest, the most seductive, the most effective instrument of moral propaganda in the world, excepting only the example of personal conduct; and I waive even this exception in favor of the art of the stage, because it works by exhibiting examples of personal conduct made intelligible and moving to crowds of unobservant unreflecting people to whom real life means nothing.”
—George Bernard Shaw (1856–1950)
““You, that have not lived in thought but deed,
Can have the purity of a natural force,
But I, whose virtues are the definitions
Of the analytic mind, can neither close
The eye of the mind nor keep my tongue from speech.””
—William Butler Yeats (1865–1939)