I have a problem understanding the Weierstrass M test and its connection with uniform convergence. A series proved to be uniform convergent by the M-test has to be continous by the properties of uniform convergence, however I find myself confused by this as I do not understand how this can be. For example consider the sum G_n(x) from 0 to infinity, in an arbitary intervall, where I define all G_n to be 0, except G_0 which I define to be equal to 0 at x=0 and equal to 1 otherwise.
Surely this pass the M-test? But yet it is not continous? Where am I going wrong?