Let #f(x)# be a function defined in an interval #(a,b)# and #x_0 in (a,b)# a point of the interval.

Then the definition of continuity is that the limit of #f(x)# as #x# approaches #x_0# equals the value of #f(x)# in #x_0#.

In symbols:

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

Based on the formal definition of limit, then, for every number #epsilon > 0# we can find #delta_epsilon > 0# such that:

#abs(x-x_0) < delta_epsilon => abs(f(x)-f(x_0)) < epsilon#

This means that as #x# gets closer and closer to #x_0# also #f(x)# gets closer and closer to #f(x_0)# and thus the function is "smooth".