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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.


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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. – Gunnar Þór Magnússon May 12 '12 at 21:55
up vote 5 down vote accepted

$\lim\limits_{h\searrow a}f(h)$ 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.

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why is it written that way? I have seen that written with + or - before, to show left or right. Does it have to with style format of various branches of math? – yiyi May 12 '12 at 6:48
@MaoYiyi: I suspect that it does correlate somewhat with the branch of mathematics in which the writer works. I'm more accustomed to seeing the arrow notation for sequences, where $\langle x_n:n\in\Bbb N\rangle\searrow x$, for instance, means that $\langle x_n:n\in\Bbb N\rangle$ is a monotonically decreasing sequence converging to $x$. – Brian M. Scott May 12 '12 at 6:50
thanks, which there wasn't so much differences in various way to write the same thing in math. – yiyi May 12 '12 at 6:52
I must admit the notation is pretty much standard in semi-differential theory for functions $f : \mathbb R^n \to \mathbb R$, where one desires to evaluate the derivative fraction $$ \frac{f(x+tv) - f(x)}{t} $$ at $x,v \in \mathbb R^n$ and $t \in \mathbb R$ when $t$ decreases towards zero, i.e. you don't take the line going through $v$, just the "positive direction part of the line". It's very useful for characterizing minimums of non-differentiable functions that are semi-differentiable in every direction $v$. – Patrick Da Silva May 12 '12 at 8:12
For instance, as a simple example, the function $f : \mathbb R \to \mathbb R$ with $f(x) = |x|$. – Patrick Da Silva May 12 '12 at 8:13

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