derivative of limit function diverges from limit of derivatives

For a function sequence, one cannot always change the of limit and differentiating (, i.e. it may well be


in the case that a sequence of continuousMathworldPlanetmathPlanetmath (and differentiableMathworldPlanetmathPlanetmath) functions converges uniformly; cf. Theorem 2 of the parent entry (

Example.  The function sequence

fn(x):=j=1nx3(1+x2)j=x-x(1+x2)n  (n= 1, 2, 3,) (1)

provides an instance; we consider it on the interval[-1, 1].  It’s a question of partial sum the converging geometric seriesMathworldPlanetmath


(although one cannot use Weierstrass’ criterion of uniform convergenceMathworldPlanetmath).  Since the limit function is

f(x):=limn(x-x(1+x2)n)=xx[-1, 1],

we have

sup[-1, 1]|fn(x)-f(x)|=sup[-1, 1]|x|(1+x2)n0asn,

which means by Theorem 1 of the parent entry ( that the sequence (1) converges uniformly on the interval to the identity function.  Further, the members of the sequence are continuous and differentiable.  Furthermore,

fn(x)= 1-1+(1-2n)x2(1+x2)n+1,


limnfn(x)= 1(x 0).

But in the point  x=0  we have

limnfn(0)=limn0= 0,

which says that the limit of derivativePlanetmathPlanetmath sequence of (1) is discontinuousMathworldPlanetmath in the origin.  Because

f(x) 1,

we may write

Title derivative of limit function diverges from limit of derivatives
Canonical name DerivativeOfLimitFunctionDivergesFromLimitOfDerivatives
Date of creation 2013-03-22 19:00:29
Last modified on 2013-03-22 19:00:29
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 40A30
Classification msc 26A15
Synonym limit of derivatives diverges from derivative of limit function
Related topic GrowthOfExponentialFunction