Thread: Definition of a Continuous function

Is the epsolon-delta definition of a continuous function stating that however small a neighbourhood around 'f(c)' we want f(x) to be, we can choose a sigma such that if the distance between x and c is less than sigma, the former is implied. Ok not very well expressed but you know what I mean. Then I recognised that this is simply the definition of a limit when L=f(c). So what is the definition reffering to. The limit, or the the intuitive notion that small changes in x produce small changes in f(x). I feel there is a cause and effect idea I'm missing. Thanks

There are two distinct differences between those definitions.

First: for the limit definition, x is not allowed to reach c. That is, you have



For the continuity definition, you relax the condition so that x can actually equal c:



Second: as you've already pointed out, with the limit definition, no attempt is made to provide any indication as to what the limit is. It's just an unknown number L. Whereas, in the continuity definition, it must be f(c).

Thanks. I am still unsure as to whether the epsilon-delta definition of continuity is reffering to F(x) tending to F(c) as x tends to c or is it as wikipedia says Continuous function - Wikipedia, the free encyclopedia in this paragraph; Weierstrass definition (epsilon-delta) of continuous functions. Or is it both. I suppose they are equivalent statements

The statement that the limit as x goes to c of f(x) equals f(c) is entirely equivalent to the Weierstrass definition of continuity. The first is what we call the "calculus" definition, often.

Ok thanks, I find the world of analysis very interesting indeed.

You're welcome! I would agree that analysis is very interesting.