# What's the definition of this limit?

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:

## 1 Answer

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).

• 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. – Ashwin B Jan 15 at 1:53
• 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.) – Joel Pereira Jan 16 at 1:02