Suppose X = [0,1] x [0,1] and d is the metric on X induced from the Euclidean metric on R^2. Suppose also that Y = R^2 and d' is the Euclidean metric. Is the mapping
T: [0,1] x [0,1] [tex]\rightarrow[/tex] R^2, T(x,y) = (xy, e^(x.y))
uniformly continuous? Explain your answer.
The Attempt at a Solution
So I know the definition for uniformly continuous, but am wondering if it's necessary to use it? We have in our notes that continuous linear maps on normed vecotr spaces are unifomrly continuous, and (Y,d') is a normed vector space.
So by looking at the graph of the map, there is a discontinuity between the line on the x-axis and the exponential function. So can you say it is not continuous and thus not uniformly continuous?
Thanks for any help