Interchange of limit and maximum of a function?

[Bahareh Akhbari]

Hi All

Let f(x,y) is a two-variable function. Can we write:

lim_{x->0} ( max_{y} f(x,y) ) = max_{y} ( lim_{x->0} f(x,y) ),

assuming that all limits and maximums exist and are finite?

I need it especially for the case f(x,y) is continuous but lim_{x->0}
f(x,y) is not necessarily a continuous function of y.

Thanks in advance for your comments

[Arturo Magidin]

On Mar 26, 10:32 pm, Bahareh Akhbari <akhbari.baha...@gmail.com>
wrote:

> Hi All
>
>    Let f(x,y) is a two-variable function. Can we write:
>
> lim_{x->0} ( max_{y}  f(x,y) ) = max_{y}  ( lim_{x->0}  f(x,y) ),
>
> assuming that all limits and maximums exist and are finite?

Take f(x,y) = sin(xy).

For any x=/=0, we have that max_y (f(x,y)) = max_y sin(xy) = 1, since
we can always choose y so that sin(xy) = 1. So

lim_{x->0} max_y f(x,y) = 1.

However, for any fixed y, lim_{x->0} f(x,y) = 0, so

max_y (lim_{x->0} sin(xy)) = max_y (0) = 0.

--
Arturo Magidin