Uniform convergence of piecewise continuous functions

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SUMMARY

The discussion centers on the uniform convergence of sequences of piecewise continuous functions, specifically on the question of whether such sequences can converge uniformly to a function that is not piecewise continuous. Participants highlight that a sequence of continuous functions converging uniformly results in a continuous limit, suggesting that the piecewise nature of the functions does not alter this outcome. The mention of Fourier series indicates a potential avenue for exploration, although clarity on the convergence behavior remains elusive.

PREREQUISITES
  • Understanding of uniform convergence in real analysis
  • Familiarity with piecewise continuous functions
  • Knowledge of Fourier series and their properties
  • Basic concepts of limits and continuity in mathematical analysis
NEXT STEPS
  • Explore the properties of uniform convergence in the context of piecewise functions
  • Investigate the convergence of Fourier series and their limits
  • Study examples of functions that are continuous but not piecewise continuous
  • Examine the implications of uniform convergence on the continuity of limits
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Students and professionals in mathematics, particularly those studying real analysis, as well as educators seeking to deepen their understanding of convergence concepts and their applications.

Demon117
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I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous?

I've thought of letting each of these sequences of functions being piecewise in terms of the Fourier series but I am unsure this really works because I haven't figured out what it would converge to.

Any suggestions or other examples?
 
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A sequence of continuous functions which converges uniformly has a continuous limit. I do not see how piecewise would change anything. As all sequence functions are joined at the same point, the limit will be, too.
 

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