The function #f(x) = x^5# is defined at #x = 0# and for any numbers #x_1 < x_2#, we have #f(x_1) < f(x_2)#.
Examples for comparison
The function #g(x) = -1/x# is not defined at #0# and althoug the derivative is always positive where it is defined, the function is not increasing on #(-oo,oo)#. It is increasing on each of #(-oo,0)# and #(0,oo)#.
Graph of #y = -1/x#
graph{-1/x [-14.24, 14.24, -7.12, 7.12]}
The function #h(x) = {(x,"if",x < 0),(x-3,"if",x >= 0):}# is also increasing on each of #(-oo,0)# and #(0,oo)# but is not increasing on #(-oo,oo)#
