8.1. Pointwise Convergence

Example 8.1.8 (d): Pointwise Convergence does not inherit Derivatives

Find a pointwise convergent sequence of differentiable functions whose limit is differentiable but the sequence of derivatives does not converge.


fn(x) = 1/n sin(n x)

Then | fn(x) | 1/n for all x so that the sequence converges to zero for all x. Each fn(x) is differentiable with

f 'n(x) = cos(n x)


  • The sequence fn(x) converges pointwise to zero
  • Each fn(x) is differentiable
  • The limit function is differentiable
  • The sequence of derivatives does not converge

Oops, I guess we should justify our claim that f 'n(x) = cos(n x) does not converge pointwise. Can you? As a hint, it would be enough to find one fixed x for which the resulting numeric sequence diverges.

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