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I am reading Amann and Escher's trilogy on Analysis and at the very beginning of volume II, the authors begin with the definition of a step functions and what it means for a step function to be "jump continuous". Intuitively, I know what this means but the authors use very unfamiliar notation at this point. Specifically, they use $f(a + 0)$ to denote the limit of $f$ as $x$ approaches $a$ from the right (at least, this is what I think they mean by it) I have rewritten the definition as I understand it using notation that is more familiar and it reads:

A function $f:I\rightarrow E$ from the perfect interval I to the Banach space $(E, ||\cdot||)$ is called a step function if $I$ has a partition $\mathcal{P} = (\alpha_0, \dots, \alpha_n)$ such that $f$ is constant on every open interval $(\alpha_{j-1},\dots \alpha_j)$ Moreover, if the limits $\lim \limits_{x \to \alpha^+}{f(x)}$ and $\lim \limits_{x \to \beta^-}{f(x)}$ exist and the limits $\lim \limits_{x \to a^-}{f(x)}$ and $\lim \limits_{x \to a^+}{f(x)}$ exist for each $a$ in the interior of $I$ then $f$ is said to be jump continuous

Can anyone comment on the correctness of the above definition and verify that I have interpreted the authors' meaning correctly?

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    If you mean $I = [\alpha,\beta] = [\alpha_0,\alpha_n]$, then yes, you interpreted it correctly. The point of jump continuous functions is to distinguish it from functions which "blow up" at some points: the Heaviside function restricted to $[-1,1]$ is jump continuous, but $1/x$ restricted to the same interval is not.2011-05-04
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    Yes, that is the interval in question. Sorry I wasn't more precise. Thanks for the clarification/confirmation2011-05-04

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There isn't much more to say to answer this question, except to confirm that you understood the definition correctly. I also find the notation $f(a+0)$ for the limit from one side of $a$ of the function $f$ quite strange.

Step functions are fundamental for various reasons; among the most elementary is that they are typically used in the development of the Lebesgue integral, although they (well, their `derivatives') are also typically used to motivate the theory of distributions and generalised functions.

Perhaps one can also store a link to the Wikipedia page here.

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    What is even more strange is that I cannot find that notation defined anywhere else in the text, including Volume I. It's certainly possible however that I've overlooked it. The orginal text though is in German; as odd as it sounds, mabye that notation is more common in German-authored texts?2011-05-06
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    I don't think so. As chance would have it, I'm working as a mathematician in Germany at the moment and nobody I know uses that notation!2011-05-06
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    I've actually encountered another text which uses this bizarre notation: Suhubi's Functional Analysis, c.f., page 178-1792011-07-13