The discussed 2D function f(x,y) = a + bx + cy + dxy does not have an inverse because it maps multiple input pairs to the same output, violating the definition of a function. An example provided illustrates that f(0,0) and f(0,1) both yield the same result, making it impossible to define a unique inverse. Generally, functions from R^2 to R cannot have continuous inverses, as this would imply a homeomorphism between the plane and a line, which is not feasible. In contrast, functions mapping from R^2 to R^2 might have continuous inverses, but this is contingent on the specific function. Therefore, the original function lacks an inverse due to its inherent properties.
