I've only dealt with the following definition of limits:

For a function $f:E\to \Bbb R, E \subseteq\Bbb R$ $$\lim \limits_{E\owns x\to a} f(x)= L :=\forall \epsilon >0\ \exists\ \delta >0\ \forall x\in E\ \Bigl(0<|x-a|<\delta \implies |f(x)-L|<\epsilon \Bigr)$$

The material in the reference is from a book (Signals and Systems 2nd Edition, Oppenheim) and it makes the following statement:

$$u(t) = \lim \limits_{\Delta \to 0} u_\Delta (t)$$

Questions on the notation $u_\Delta(t)$:

I've been told that if a function $f$ has a domain $D$ and a codomain $C$ you must declare it as $f:D \to C$ and use the function value at point $x \in D$ as $f(x)$. What does it means when the independent variable is a part of the function symbol (as in the case of $u_\Delta$, where I'm assuming $\Delta$ is an independent variable)? Is this some alternate syntax for functions of several variables? If that's so could you provide me the definition of the syntax?

Question on the limit definition:

What is the definition of $\lim \limits_{\Delta \to 0} u_\Delta (t)$?


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Your definition is that of a limit of a function at a particular point. In your reading, you're looking at the limit of a sequence of functions, in this case point wise convergence. That is, we have for every $\Delta$, a function u$_\Delta$. Then we define u(t) = $\displaystyle\lim_{\Delta\rightarrow0}u_{\Delta}$(t).

Graphically, we see what happens to the values of u$_{\Delta}$(t) as $\Delta$ changes. Whatever value u$_\Delta$(t) tends to, we define that as u(t).

  • $\begingroup$ But does ud(t) really converge to u(t) pointwise? What analysis should I study to understand convergence of a sequence of functions? So far I've only studied real analysis. $\endgroup$ – Ashwin B Jan 15 '19 at 1:53
  • $\begingroup$ There are different type of convergence involving functions. These can be found in any real analysis text. There's pointwise convergence, uniform convergence (which preserves continuity and differentiability.) $\endgroup$ – Joel Pereira Jan 16 '19 at 1:02

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