Given the function #f(x) = 0#, since this is a constant function (that is, for any value of #x#, #f(x) = 0#, the limit of the function as #x->a#, where #a# is any real number, is equal to #0#.
More specifically, as a constant function, #f(x)# maintains the same value for any #x#. #f(1) = f(2) = f(pi) = f(e) = f(-1258567/44) = 0#. Further, as a constant function, #f(x)# is by definition continuous throughout its domain. Note, however, that if one arrives at a constant function via division or multiplication of non-constant functions (for example, #(8x)/x#, there will still exist a discontinuity where the original denominator was #0#).
Graphing the function will further prove this point. On the graph #f(x) = 0#, the curve is simply the #x#-axis, which maintains a constant #y#-value of #0# no matter the #x#-value.
