6
$\begingroup$

I was reading "A Course in Probability Theory" by Kai Lai Chung, and in the book he was discussing discontinuity of monotonic functions, and after doing some searching online to learn more about the various concepts of discontinuity, I stumbled across Froda's Theorem and I have no idea what the arrows mean in part 2 of the definition:

$\qquad\qquad\Large f(x+0):=\lim\limits_{h\searrow \,0} f(x+h)\quad$ and $\quad\Large f(x-0):=\lim\limits_{h\nearrow \,0} f(x-h)$

I have never seen those symbols before. I have a basic idea of the proof listed on wikipedia, and I am still working on fully understanding it; however, I can't find what those arrows mean.

Thanks.

  • 0
    This is both the first time I see this slanted arrow notation, and the first time in ten years I don't have to think before knowing from where we approach the limit.2012-05-12

1 Answers 1

6

$\lim\limits_{h\searrow a}f(h)$ means the same as $\lim\limits_{h\to a^+}f(h)$: it's the limit of $f(h)$ as $h$ approaches $a$ from the right. Similarly, $\lim\limits_{h\nearrow a}f(h)$ means the same as $\lim\limits_{h\to a^-}f(h)$: it's the limit of $f(h)$ as $h$ approaches $a$ from the left. Since bigger numbers are on the right, approaching $a$ from the right can also be thought of as approaching $a$ from above, i.e., from higher numbers. Similarly, approaching $a$ from the left can be thought of as approaching $a$ from below, i.e., from smaller numbers.

  • 0
    Which way you write these one-sided limits probably depends on who your teacher was, or what country your textbook was from.2012-05-12