Thread: question on continuity of functions

Okay, so in general, if I wanted to do a proof to show that some function f(x) is continuous for all x, how do i do that? I mean, general proofs of continuity are easy, but how do you prove it to be continuous for all x in a set?

Originally Posted by jmoney90

Okay, so in general, if I wanted to do a proof to show that some function f(x) is continuous for all x, how do i do that? I mean, general proofs of continuity are easy, but how do you prove it to be continuous for all x in a set?

Choose any point in the set, say .
Then by showing that the function is continuous at suffices to show that is continuous on the set.

Originally Posted by Plato

Choose any point in the set, say .
Then by showing that the function is continuous at suffices to show that is continuous on the set.

Okay, thanks. So just to clarify, basically you prove it by being vague and general, and not picking specific values?

Originally Posted by jmoney90

So just to clarify, basically you prove it by being vague and general, and not picking specific values?

Why do you think that there is anything vague about that method?

You suppose that and show that is continious at .
There is absolutely nothing vague about that.

Maybe you don't understand the definition of continuity. Is that it?

Originally Posted by Plato

Why do you think that there is anything vague about that method?

You suppose that and show that is continious at .
There is absolutely nothing vague about that.

Maybe you don't understand the definition of continuity. Is that it?

I guess general was more the word I was looking for. I understand the idea behind continuity: the limit as x approaches some point is the same as that function at some value, and alos how to formally work the definition. The book didn't give any examples of a prooof in which you'd prove continuity of a function in general.