Calculus: Understanding Lower/Upper Bounds and Epsilon

[The_ArtofScience]

I recently opened a calculus textbook to understand the exact definitions of lower upper bound and greater lower bound. The book went into a discussion over "neighborhoods" explaining that if one were asked to prove a limit point, assume that x lies between 0 and a funny looking symbol that looks like some curled S. To prove lower upper bounds and greater lower bounds, the symbol episolon was used.

I'm curious, what is the real difference between episolon and that curled S? (Sorry I couldn't find a latex image for it but it looks like partial charges from chemistry)

 

[LukeD]

You mean $$\delta$$? That's a delta, it's Greek the version of a d. The capital version looks like this: $$\Delta$$

Assuming that you did mean delta, the difference is that they are typically used in different settings. For instance, the definition of a continuous function is this:

For all $$\varepsilon > 0$$ and for all $$x, \ \exists \delta > 0$$ such that if $$\lvert x - y \rvert < \delta$$ then $$\lvert f(x) - f(y) \rvert < \varepsilon$$

Epsilon is some arbitrary but fixed value such that the statement is true for all positive values.
Delta is some selected value that makes the statement work for a specific epsilon and for a specific x.

In general, the situation is which you will see them is that epsilon is arbitrary but fixed (and you want your proof to work for all values of epsilon > 0) whereas delta is going to be some particular number (depending on the situation, it might depend on some other numbers such as epsilon or x) that specifies some sort of bounded interval that makes the proof work.

 

[heybabylover]

Funny

hey, man. May i have a question? May i post a message to you? [/size]