Need an example of a sequence of functions that is discountinuous at every point on [0,1] but converges uniformly to a function that is continuous at every point
The Attempt at a Solution
I used the dirichlet's function as the template
f_n(x) = 1/n if x is rational and 0 if x is irrational
f_n(x) is discontinuous at every x in [0,1] and converges to f(x)=0
But this seems to be a erroneous analysis, because 1/n eventually goes to 0 so f_n(x) will be continuous as n->infinity
Can i get help in constructing this?