How do you prove that the function #f(x) = x^(1/2)# is continuous at x=1/2?
1 Answer
There are many ways to prove this  one could use :

Epsilondelta methods
#y=f(x)# is continuous at#x=x_0iff AA epsilon >0, EE delta>0# such that#xx_0 < delta=> f(x)f(x_0) < epsilon# 
Topological methods
#y:X>Y;y=f(x)# is continuous everywhere (specifically at#x_0 in X# iff and only if for every open subset A in Y, the inverse function#f^(1)(A)# is open in X. 
Functional analysis methods using sequential criterion .
#y=f(x)# is continuous at#x=x_0iff# for every sequence#(x_n)# converging to#x_0# , the sequence of function values#(f(x_n))# converges to#f(x_0)# .
I shall prove the continuity using the latter definition in this case :
Let
Since
But