Question

Not actually asking about this question, but rather the jargon used:
“If f(x) is continuous at c, does that mean that computing its limit at c is not worthwhile?"
is it different from: “If YOU KNOW THAT f(x) is continuous at c, does that mean that computing its limit as c is not worthwhile?”

Answer 1

@Hero

Answer 2

Those two statements are equivalent.

Answer 3

My math professor argues otherwise.

Answer 4

I would agree with you.

Answer 5

saying "If f(x) is continuous at c" is implying that f(x) is continuous at c. right?

Answer 6

Yes. Technically, you are assuming f(x) is continuous at c. I suppose you don't actually "know" it is. But, in terms of logic, they are equivalent statements.

Answer 7

Standing from a mathematical viewpoint rather than an english language viewpoint, would they be different?

Answer 8

No. The other way around - they are equivalent from a mathematical (logical) standpoint. In terms of English, they could be considered different.

Why does your professor think otherwise?

Answer 9

He thinks that if the question states, "if f(x) is continuous", then we actually don't know if it is continuous or not.

Answer 10

Unless you justify something, you cannot know if its true. But, assuming it means that the justification has been given previously and omitted (usually, anyway).

I don't think your professor has his head on straight, but who knows. Maybe I'm not seeing it. This "difference" has never hindered me from doing valid proofs.

Answer 11

Also, to add further, there may actually be a difference and I've just never had to deal with it. Mathematicians are pretty lazy when it comes to writing statements. So, most people would shorten that second statement into the first, by removing "you know that".

Answer 12

okay, thanks for the help.

Answer 13

Actually, I'm going to retract my statement. I think I see it now. From a mathematical standpoint, they're usually considered equivalent.

From a logical standpoint, the first statement gives you this:
if p, then q

The second statement gives you this:
if p is true and if p, then q

When doing a mathematical proof, you (usually) start with "suppose p is true/let p be true/etc.". This allows you to do the proof.

Does that make sense?

Answer 14

@satellite73 @campbell_st
any opinion?

Answer 15

@DanielxAK that does make sense, but this is still very controversial in my mind.