because all the positive terms for are added. Therefore, dividing this inequality by and taking the limit from above,
We now prove the formula for the nth derivative of f by mathematical induction. Using the chain rule, the reciprocal rule, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of f for all x > 0 and that p1(x) is a polynomial of degree 0. Of course, the derivative of f is zero for x < 0.
It remains to show that the right-hand side derivative of f at x = 0 is zero. Using the above limit, we see that
The induction step from n to n + 1 is similar. For x > 0 we get for the derivative
where pn+1(x) is a polynomial of degree n = (n + 1) − 1. Of course, the (n + 1)st derivative of f is zero for x < 0. For the right-hand side derivative of f (n) at x = 0 we obtain with the above limit
has a strictly positive denominator everywhere on the real line, hence g is also smooth. Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the unit interval [0, 1]. To have the smooth transition in the real interval [a, b] with a < b, consider the function
For real numbers a < b < c < d, the smooth function
equals 1 on the closed interval [b, c] and vanishes outside the open interval (a, d), hence it can serve as a bump function.
A smooth function which is nowhere real analytic
A more pathological example is an infinitely differentiable function which is not analytic at any point. It can be constructed by means of a Fourier series as follows. Define for all
Since the series converges for all , this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test to demonstrate uniform convergence of each series of derivatives.
We now show that is not analytic at any dyadic rational multiple of π, that is, at any with and . Since the sum of the first terms is analytic, we need only consider , the sum of the terms with . For all orders of derivation with , and we have
where we used the fact that for all , and we bounded the first sum from below by the term with . As a consequence, at any such
For every sequence α0, α1, α2, . . . of real or complex numbers, the following construction shows the existence of a smooth function F on the real line which has these numbers as derivatives at the origin. In particular, every sequence of numbers can appear as the coefficients of the Taylor series of a smooth function. This result is known as Borel's lemma, after Émile Borel.
With the smooth transition function g as above, define
This function h is also smooth; it equals 1 on the closed interval [−1,1] and vanishes outside the open interval (−2,2). Using h, define for every natural number n (including zero) the smooth function
which agrees with the monomialxn on [−1,1] and vanishes outside the interval (−2,2). Hence, the k-th derivative of ψn at the origin satisfies
and the boundedness theorem implies that ψn and every derivative of ψn is bounded. Therefore, the constants
involving the supremum norm of ψn and its first n derivatives, are well-defined real numbers. Define the scaled functions
This pathology cannot occur with differentiable functions of a complex variable rather than of a real variable. Indeed, all holomorphic functions are analytic, so that the failure of the function f defined in this article to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.
Note that although the function f has derivatives of all orders over the real line, the analytic continuation of f from the positive half-line x > 0 to the complex plane, that is, the function
has an essential singularity at the origin, and hence is not even continuous, much less analytic. By the great Picard theorem, it attains every complex value (with the exception of zero) infinitely many times in every neighbourhood of the origin.
^See e.g. Chapter V, Section 2, Theorem 2.8 and Corollary 2.9 about the differentiability of the limits of sequences of functions in Amann, Herbert; Escher, Joachim (2005), Analysis I, Basel: Birkhäuser Verlag, pp. 373–374, ISBN3-7643-7153-6