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?
Answer
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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