1
$\begingroup$

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

REFERENCE:

enter image description here enter image description here

$\endgroup$
1
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ 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 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 at 1:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.