Continuity of series of functions #
We show that series of functions are continuous when each individual function in the series is and
additionally suitable uniform summable bounds are satisfied, in continuous_tsum
.
For smoothness of series of functions, see the file Analysis.Calculus.SmoothSeries
.
An infinite sum of functions with summable sup norm is the uniform limit of its partial sums. Version relative to a set, with general index set.
An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
Version relative to a set, with index set ℕ
.
An infinite sum of functions with summable sup norm is the uniform limit of its partial sums. Version with general index set.
An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
Version with index set ℕ
.
An infinite sum of functions with summable sup norm is continuous on a set if each individual function is.
An infinite sum of functions with summable sup norm is continuous if each individual function is.