Why is the continuous function $ x^{2} + 2x - 7 $ on $ (1,4) $ uniformly continuous?

Question

Most of the other related questions, asking if a set was uniformly continuous we're easily answered by the fact that a set was either compact (so it was uniformly continuous) or it wasn't (so it wasn't uniformly continuous).

But I can't really figure out why this function is supposed to be uniformly continuous then, is there anyone who can clear this out for me?

Answer 1

It's not clear to me whether you are just writing it badly or if you really have basic misunderstandings about "uniform continuity".

""Most of the other related questions, asking if a set was uniformly continuous we're easily answered by the fact that a set was either compact (so it was uniformly continuous) or it wasn't (so it wasn't uniformly continuous). "

A set is not "uniformly continuous"- you mean a function on a given set. And the rest of that is not correct. What is true is "If a function is continuous on a compact set then it is uniformly continuous on that set". That does NOT say that a continuous function is not uniformly continuous on a set that is NOT compact- it may or may not be.

However, there is another important property of "uniformly continuous" functions is "If f is uniformly continuous on a set, A, it is uniformly continuous on any subset of A".

Here, as MathematicsStudent1122 said, this function is continuous on the closed and bounded, so compact, set [1, 2] so is uniformly continuous on that set. Therefore, it is uniformly continuous on (1, 2), a subset of [1, 2].

Answer 2

A direct proof: Let $f(x)=x^2+2x-7$. For $x \in (1,4)$ we have

$|f(x)-f(y)|=|(x+y)(x-y)+2(x-y)| \le |x-y|(2+|x|+|y|) \le 10|x-y|$.