Switching Limits and also Suprema

intend, $f(x,y)$ is a bounded continual function on $\mathbb{R}^2$. Take into consideration $$\lim_{ y \rightarrow y' }\> \sup_{x \in \mathbb{R}} f(x,y).$$

In just how much can you switch over suprema and also restrictions, or which feasible term comes closest to switching over these 2? Preferably, there would certainly be something like $$\sup_{x \in \mathbb{R}}\> \lim_{ y \rightarrow y' } f(x,y).$$


2019-12-02 02:54:01
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Answers: 1

You can not as a whole. To take a really straightforward instance, take into consideration $f(x,y) = \sin(xy)$, and also $y'=0$. After that we have : $$\lim_{y\to 0}\>\sup_{x\in\mathbb{R}}\sin(xy) = 1.$$ (Remember that if we take the restriction as $y\to 0$, after that we do rule out $y=0$, so $y\neq 0$ in $\sin(xy)$). Yet $$\sup_{x\in\mathbb{R}}\>\lim_{y\to 0}\sin(xy) = \sup_{x\in\mathbb{R}}\ 0 = 0.$$

However, if the restriction of $f(x,y)$ as $y\to y'$ exists for every single $x$, and also the restriction of the suprema exist, after that you get one inequality : $$\sup_{x\in\mathbb{R}}\>\lim_{y\to y'}f(x,y) \leq \lim_{y\to y'}\>\sup_{x\in\mathbb{R}} f(x,y).$$ To see this, keep in mind that for each and every dealt with $x_0\in\mathbb{R}$, $f(x_0,y) \leq \sup\limits_{x\in\mathbb{R}}f(x,y)$, so taking limits we have $$\lim_{y\to y'} f(x_0,y) \leq \lim_{y\to y'}\>\sup_{x\in\mathbb{R}}f(x,y).$$ Since this holds for each and every $x_0$, the supremum additionally pleases the inequality, so $$\sup_{x\in\mathbb{R}}\>\lim_{y\to y'}f(x,y) \leq \lim_{y\to y'}\>\sup_{x\in\mathbb{R}}f(x,y).$$

2019-12-03 04:22:26