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I have troubles understanding the definitions for right-hand/left-hand jump discontinuities.

As given in e.g.

https://www.diva-portal.org/smash/get/diva2:5850/FULLTEXT01.pdf

$f:[a,b] \rightarrow \mathbb{R}$ monotone on $[a,b]$, $c \in [a,b]$.

right-hand jump:

given $c \in [a,b)$

$$\sigma_c^+=f(c+0)-f(c)$$

left-hand jump:

given $c \in (a,b]$

$$\sigma_c^-=f(c)-f(c-0)$$

These are intuitively $=0$? But then the author does a proof where he claims that for $f$ increasing these would be positive?

What's the meaning of $f(c+0)$ or $f(c-0)$? Aren't these same as $f(c)$?

mavavilj
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    About your last question, look at the bottom of the 3rd page of the pdf. $f(c+0)$ is not $f(c)$, it's a notation for $\lim_{h\to 0} f(c+h)$. – Al.G. Sep 09 '20 at 10:46
  • @Al.G.Yes this clears it. The notation is perhaps a bit confusing. – mavavilj Sep 09 '20 at 11:05

2 Answers2

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The notations $f(c+0)$ and $f(c-0)$ are first introduced in Theorem 1.0.2, which says (in part)

Then $f(c+ 0)$ and $f(c−0)$ $\color{red}{^1}$ exists

where I have colored the footnote number $\color{red}{1}$ red for emphasis. If you look at footnote number $1$ at the bottom of the same page, you may see that it says

$^1$ We denote the right-hand and left-hand limits $$\lim_{h\to0}f(c+h) =f(c+ 0) \qquad \lim_{h\to0}f(c−h) =f(c−0)$$ where $h$ tends to $0$ from the positive side.

So these somewhat unusual notations are related to the function $f$ but are not the same as the function $f.$ You can mentally substitute the left-hand side of each of these definitions whenever you see the right-hand side.

David K
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If $f$ is increasing, then $\sigma_c^+ \ge 0$ and $\sigma_c^- \ge 0$.

We have $\sigma_c^+ = 0 \iff \lim_{x \to c+0}=f(c)$ and $\sigma_c^- = 0 \iff \lim_{x \to c-0}=f(c)$.

Fred
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