a function that has an asymptote is uniformly continuous

Let be a continuous function and . Show that is uniformly continuous.

I'd like to use this charactetization of uniform continuity. A function is uniformly continuous iff for all sequences if , then also .

Suppose .

If is a bounded sequence, then must be bounded too, and so if , then also , because is continuous and therefore uniformly continuous on a compact set. If, on the other hand, , then also and we have

But what if is neither bounded nor convergent to infinity?

Re: a function that has an asymptote is uniformly continuous

Quote:

Originally Posted by

**ymar** Let

be a continuous function and

. Show that

is uniformly continuous.

I'd like to use this charactetization of uniform continuity. A function

is uniformly continuous iff for all sequences

if

, then also

.

Suppose

.

If

is a bounded sequence, then

must be bounded too, and so if

, then also

, because

is continuous and therefore uniformly continuous on a compact set. If, on the other hand,

, then also

and we have

But what if

is neither bounded nor convergent to infinity?

Let and . Then there is such that if then . Notice also that is uniformly continuous on (by Heine-Cantor), where . So there is such that if for then . Now let , and suppose . If then (details omitted). Otherwise or , giving us and hence . So in both cases we have implying .