Show there is no continuous injective maps : R^2----> R.
Suppose there is.
We know that is homeomorphic to the two dimensional sphere minus a point, , and that is homeomorphic to the circle minus a point, .
So we have a continuous and injective map .
Remove a point from .
Choose two points , such that is contained in the arc to join and . There exists a simple curve on to connect . Then, must be a continuous simple curve to connect on . This is a contradiction as belong to two disjoint connected components of .