8.1. Pointwise Convergence

Example 8.1.8 (a): Pointwise Convergence does not preserve Continuity

Find a pointwise convergent sequence of functions, each of which is continuous, but whose limit function is not continuous.

Let fn(x) = xn for 0 [0, 1] and define


  • each fn(x) is continuous (even differentiable)
  • fn(x) f pointwise
  • the limit function f is not continuous at x=1

Thus, pointwise convergence does not preserve continuity.

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