Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y0 in Y, the map h: X-> Z defined by h(X)= F( x x y0) is continuous, and for each x0 in X, the map k: Y-> Z defined by k(y) =F(x0 x y) is continuous. Show that if F is continuous, then F is continuous in each variable separately.
I'm not sure how to do this....
if you guys would help me out that would be amazing!
Suppose F(x,y) is continuous at (a,b) then we must prove h(x) is continuous at a and k(y) is continuous at b.
Originally Posted by tomboi03
If F(x,y) is continuous at (a,b) then given ε>0 there exists δ>0,such that
|F(x,y)-F(a,b)|<ε ,for all x,y satisfying |x-a|<δ,|y-b|<δ.
Since |b-b|=0<δ. By putting y=b in the above inequalities we have:
|F(x,b)-F(a,b)|<ε ,for all x satisfying |x-a|<δ.
But by definition F(x,b)=h(x),and F(a,b)=h(a), so the above inequalities become:
|h(x)-h(a)|<ε,for all x satisfying |x-a|<δ.
Hence h(x) is continuous at a.
In exactly the same way we prove k(y) is continuous at b.
The converse of the above theorem is not necessarily true