Is there a summation rule for continuous functions?

The sum of two continuous functions is continuous.

Let #f(x)# and #g(x)# be two real functions of real variable defined and continuous in #(a,b)#. If #x_0 in (a,b)#:

#lim_(x->x_0) f(x)+g(x) = lim_(x->x_0) f(x) + lim_(x->x_0)g(x) =f(x_0)+g(x_0)#

and clearly the result extends to the sum of any finite sum of continuous functions.

However let #f_n(x)# with #n in NN# be a sequence of real functions of real variable defined and continuous in #(a,b)# and consider the series:

#sum_(n=0)^oo f_n(x)#

For the values of #x# for which the series is convergent we have a new function:

#f(x) = sum_(n=0)^oo f_n(x)#

that is the sum of the series. But while every partial sum is defined and continuous in #(a,b)# we cannot say the same for #f(x)# because there can be values of #x# for which the function is either not defined or not continuous.

For instance consider the functions:

#f_n(x) = cos^n(x)#

that are defined and continuous for every #x in RR#. The sum:

#f(x) = sum_(n=0)^oo cos^n(x) = 1/(1-cosx)#

that we can find based on the sum of the geometric series is not defined for #x = 2kpi#. (and not really the sum of the series for #x=kpi#).

Similarly we can demonstrate based on Fourier analysis that the series:

#sum_(n=0)^oo sin((2n+1)t)/(2n+1)#

is defined for every #x# but not continuous for #x=kpi#.

A sufficient condition for the sum of the series to be continuous, is that the series is totally convergent, that is for every #n# we can find #a_n# such that:

#abs (f_n(x)) <= a_n# for #x in (a,b)#

and that the series:

#sum_(n=0)^oo a_n #

is convergent.