Why don't you learn to use LaTeX?
As is, I for one, have no idea about what you have posted.
Let F: [0,1]X[0,1] --> R
where F is such st both Fy(x) and Fx(y) are continuous. [That is, Fy(x) is the function holding y constant]
Furthermore, assume that the set of functions, E, is equicontinuous. Where E
E = {maps that take x-->f(x,y) : y is in [0,1]}
Prove that f is continuous.
I guess my biggest problem is that I'm not entirely clear what the set E represents. Is E the set of all {Fy(x) for all y in [0,1]}?
Ok I have the answer now.
Fix e>0 and (a,b) [0,1] X [0,1]. We want to find a d such that:
l(x,y) - (a,b)l < d implies lf(x,y) - f(a,b)l < e
By equicontinuity of E, there is a d1 where
l x - a l < d1 implies lf(x,y) - f(a,y)l < e/2 for all y [0,1]
Similarly, by continuity of f(x0, y) where x0 is constant and x0 [0,1], there is a d2 where:
l y - b l < d2 implies lf(a, y) - f(a, b)l < e/2.
Let d =min{d1, d2}...