Proving Uniform Continuity of f(x): Let x in [Infinity, 0)

[minyo]

Homework Statement

let f(x)= (x^2)/(1+x) for all x in [ifinity, 0) proof that f(x) is uniformly continuous. can anyone help me with this problem

Homework Equations

using the definition of a uniform continuous function

The Attempt at a Solution

i did long division to simplify the problem and got (x-1) - 1/(x+1)

 

[HallsofIvy]

What is the definition of "uniformly continuous" and how have you tried to use it?

 

[JG89]

Find a delta that doesn't depend on any specific point in the domain of f.

 

[minyo]

for u and v in D,

|f(u)-f(v)|<epsilon if |u-v|<delta

 

[HallsofIvy]

Do you mean the domain to be [0, infinity) or (0, infinity)? Since "infinity" isn't a real number, "[infinity, 0)" doesn't make sense. I suspect it was supposed to be [0, infinity).

"for u and v in D,

|f(u)-f(v)|<epsilon if |u-v|<delta"

is much too "shorthand" to be useful here. f is uniformly continuous in D if f(x) is defined for all x in d and given epsilon> 0, there exist delta> 0 such that if |u-v|< delta, then |f(u)- f(v)|< epsilon.

The crucial point is, as JG89 said, the delta depends only on epsilon, not u or v.

|f(u)- f(v)|= |u^2/(1+u)- v^2/(1+v)|= |(u^2(1+v)- v^2(1+u))/(1+ u+ v+ uv)|
= |u^2- v^2+ uv(u- v))/(1+ u+ v+ uv)|= |(u+v- uv)/(1+u+v+uv)||u-v|.

If you can find an upper bound for that first fraction, you are home free.