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I have two questions to ask regarding uniform convergence for sequences of functions.

So I know that if a sequence ofcontinuousfunctions f_n : [a,b] -> R converge uniformly to functionf, thenfis continuous.

Is this true if "continous" is replaced with "piecewise continuous"? (I am not assuming that the sequence functions are discontinuous at the same points)

i.e. if f_n are each discontinuous at finitely many points, is the uniform limit function f discontinuous at finitely many points as well?

Also, does anyone know for what kinds of metric spaces (if any) is "pointwise convergence" for sequences of functions equivalent to "uniform convergence"?

Thanx.

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# Uniform convergence

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