Real Analysis: Continuity and Uniform Continuity

In summary, the function f(x) = (x^2)/((x^2)+1) is continuous on [0,infinity) and not uniformly continuous. This can be shown by using the definition of continuity and finding specific values for delta that make the function continuous, and then showing that there is no single delta that works for all points on the interval.
  • #1
danielkyulee
5
0
Question: Show that f(x)= (x^2)/((x^2)+1) is continuous on [0,infinity). Is it uniformly continuous?


My attempt: So I know that continuity is defined as
"given any Epsilon, and for all x contained in A, there exists delta >0 such that if y is contained in A and abs(y-x)<delta, then abs(f(x)-f(y))<Epsilon.

So i tried expanding the function, but still can not find the values for delta that make this continuous on [0,infinity). Any ideas?

Also, it is NOT uniformly continuous correct?

Thanks!
 
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  • #2
well its definitely continuous on [0,inf) - can you show that?

now for the "uniform continuity part", not as the actual proof, but I would consider the derivative - it should give you an idea of how to approach the problem

loosely speaking if a function is uniformly continuous it shouldn't have a derivative that misbehaves

consider
[tex] |\frac{f(y)-f(x)}{y-x}|[/tex]
in the limit y goes to x, this becomes the derivative...
 
  • #3
Hey, so one of the problem I had was that I could not find a way to show it was continuous. I tried to show it using the delta-epsilon definition of continuity, but couldn't figure out a way to find delta.
 
  • #4
ok so where did you get stuck?
 
  • #5
So i did absolute(f(x)-f(y))= abs( ((x^2)/((x^2)+1)) - ((y^2)/((y^2)+1)) ). I simplified this to get abs( ((x^2)-(y^2))/((x^2)+1)((y^2)+1)) ).

Now I tried to set it up so that abs ( x-y ) is less than 1 (which would be one of the minimum components of delta) so that I can try to abs( x-y ) less than something with respect to epsilon. However, I am stuck on how I can use <1 to find the deltas that make abs( f(x)-f(y) ) < E for any given E.
 
  • #6
danielkyulee said:
Question: Show that f(x)= (x^2)/((x^2)+1) is continuous on [0,infinity). Is it uniformly continuous?


My attempt: So I know that continuity is defined as
"given any Epsilon, and for all x contained in A, there exists delta >0 such that if y is contained in A and abs(y-x)<delta, then abs(f(x)-f(y))<Epsilon.

So i tried expanding the function, but still can not find the values for delta that make this continuous on [0,infinity). Any ideas?

Also, it is NOT uniformly continuous correct?

Thanks!

I believe you have your definitions backwards. A function is continuous at a point x if given epsilon there is a delta. If you pick some different point y, then given the same epsilon, you may need a different delta.

If the same delta works for a given epsilon regardless of x, that is uniform continuity.

What you wrote above is the definition of uniform continuity ... given epsilon, for any x there's a delta. The def of continuity is that given x, for any epsilon there is a delta that depends on x.
 

1. What is the definition of continuity in real analysis?

In real analysis, continuity is defined as the property of a function where small changes in the input result in small changes in the output. More formally, a function f is continuous at a point x if the limit of f as x approaches a exists and is equal to f(x).

2. How is continuity different from uniform continuity?

While both continuity and uniform continuity describe the behavior of a function, they differ in their definitions. Continuity focuses on the behavior of a function at a specific point, while uniform continuity looks at the behavior of a function over an entire interval. In other words, a function is continuous if it is continuous at every point, but a function is uniformly continuous if it exhibits a consistent rate of change over the entire interval.

3. What is the importance of continuity in real analysis?

Continuity is a fundamental concept in real analysis that allows us to understand the behavior of functions and make precise mathematical statements about their properties. It is used in many areas of mathematics, including calculus, differential equations, and topology.

4. How can I determine if a function is continuous?

To determine if a function is continuous, you can use the definition of continuity or check if it satisfies any of the theorems and properties of continuous functions. Some common methods include using the intermediate value theorem, the epsilon-delta definition of continuity, and the composition of continuous functions.

5. Can a function be continuous but not uniformly continuous?

Yes, a function can be continuous but not uniformly continuous. An example of such a function is f(x) = 1/x, which is continuous over its entire domain but fails to be uniformly continuous due to its unbounded rate of change near x = 0. In general, if a function is uniformly continuous, it must also be continuous, but the converse is not always true.

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