i was reading a proof on :

given that the power series converges on (-R,R) then it is continuous on the open interval(-R,R).

i know that i have to split it up into using triangular inequality where for all ε>0

let

δ'>o st l f(x) - f_m(x) l < ε/3

and

δ'' >0 st l f_m(x) - f_m(c) l < ε/3 ( continous at c).

im wondering, does δ' and δ'' depend on ε or x?