# Equicontinuity math problem

1. May 8, 2007

### e12514

Let f: [0,1] x [0,1] -> R be continuous and define f_y in C([0,1]) by f_y (x) = f(x,y) for each y in [0,1].

By definition of continuity of f, for all epsilon > 0 there exists delta(depending on epsilon) > 0 such that |(a,b) - (c,d)| < delta => | f((a,b) - f((c,d)) | < epsilon

If I fix b = d = y in [0,1] then I will get a statement (after a few lines)
for all epsilon > 0 there exists delta(depending on epsilon) > 0 such that, for all f_y in the set {f_y : y in [0,1]}, for all a,c in [0,1],
|(a,y) - (c,y)| < delta => | f_y (a) - f_y (c) | < epsilon

this is the definition for the set {f_y : y in [0,1]} to be uniformly equicontinuous.

Is the above argument valid?