Here is problem statement:

Show that if X is any topological space, and if two continuous maps have the property that for all , , then f is (continuously) homotopic to g.

And here is a homotopy (in our case)

A function such that and .

I don't have any trouble coming up with a function that satisfies the "deforming" property...for instance, there is a smooth function (I won't prove it, but there is, see Milnor or any other diff. top. book)

for

for

And so our homotopy could simply be

For that matter, we could simply chose

But the problem with both these functions is that they don't nessicarily satisfy , which is required if we want the range to be the n-sphere. So does anyone have any suggestions on how to impose that condition, or any other approaches to this problem that might prove fruitful?

Thanks.