Any two continuous maps from X to Y, where Y is a convex subset of R^n, are homotopic. For example, the functions f(t) = (sin2pit, cos2pit) and g(t) = (t,0) are maps from [0,1) to R^2. So these functions are homotopic. Intuitively, two functions are homotopic if one can be continuously deformed into the other. So is there really a continuous path in function space from f to g?