Proving Continuous Function: Epsilon-Delta Proof

In summary, to show that f(x)=\frac{x}{1-x^2} is a continuous function by means of an \epsilon - \delta proof, we need to find a positive delta such that for any given epsilon, the absolute value of the difference between f(x) and f(x_0) is less than epsilon whenever the absolute value of the difference between x and x_0 is less than delta. To find this delta, we manipulate the inequality involving f(x) and f(x_0) and epsilon until we get an inequality involving x and x_0. We can also use the fact that polynomials and quotients of polynomials are continuous to help us with this proof.
  • #1
latentcorpse
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how do i show [itex]f(x)=\frac{x}{1-x^2}[/itex] is a continuous function by means of an [itex]\epsilon - \delta[/itex] proof? oh and [itex]x \in (-1,1)[/itex]

so far i have said:
let [itex]\epsilon>0, \exists \delta>0 s.t. |x-x_0|< \delta[/itex]. now i need to show that [itex]|f(x)-f(x_0)|< \epsilon[/itex]. yes?

can't do the rest of it though...
 
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  • #2
You can't just say that a delta exists, you need to show that one exists. Assume that you have some positive epsilon. Now work with your inequality involving f(x) and f(x_0) and epsilon, and manipulate this until you get an inequality with x and x_0.
 
  • #3
its virtually impossible to factorise...

could i just say x is continuous and 1-x^2 is continuous and non zero so f is continuous?
 
  • #4
It depends on how you are required to show continuity. If you have to give an [itex]\epsilon - \delta[/itex] proof, then that's what you need to do. OTOH, if you have some theorems about continuity you can use, then that would be easier. For example, polynomials are continuous everywhere, and quotients of polynomials are continuous on any interval whose points don't cause the denominator to vanish (become zero). Does that help?
 
  • #5
Since f(x)= x/(1- x^2),
[tex]f(x)- f(x_0)= \frac{x}{1- x^2}- \frac{x_0}{1- x_0^2}[/tex]
[tex]= \frac{x(1-x_0^2)- x_0(1- x^2)}{(1- x^2)(1- x_0^2)}[/tex]
[tex]= \frac{x- xx_0^2- x_0- x^2x_0}{(1- x^2)(1- x_0^2)}[/tex]
[tex]= \frac{(x- x_0)- x_0x(x- x_0)}{(1- x^2)(1- x_0^2)}[/tex]
[tex]= (x- x_0)\frac{1- x_0x}{(1- x^2)(1- x_0^2)}[/tex]
Now you need to find a bound on
[tex]\frac{1- x_0x}{(1- x^2)(1- x_0^2)}[/tex]

You will, of course, want to stay away form x or [itex]x_0[/itex] being 1 or -1 so you might do this: Let a be the smaller of [itex]|1- x_0|[/itex] and [itex]|-1- x_0|[/tex] and require that [itex]|x- x_0|< a/2[/itex].
 

FAQ: Proving Continuous Function: Epsilon-Delta Proof

What is a continuous function?

A continuous function is a mathematical function that has no sudden jumps or breaks in its graph. This means that as the input of the function changes, the output changes gradually and smoothly.

What is an epsilon-delta proof?

An epsilon-delta proof is a method used in calculus to prove that a function is continuous at a specific point. It involves choosing a small value (epsilon) and finding a corresponding small interval (delta) around the point, such that all inputs within that interval will result in outputs within a certain distance from the original output.

Why is the epsilon-delta proof important?

The epsilon-delta proof is important because it provides a rigorous and precise way to prove the continuity of a function. It allows us to make a statement about the behavior of a function at a specific point, rather than just visually observing its graph.

What is the role of epsilon and delta in the epsilon-delta proof?

Epsilon and delta are both variables that represent small quantities. Epsilon represents the maximum distance that the output of the function can deviate from its original value, while delta represents the maximum distance between the input and the point in question. The goal of the proof is to find a delta that satisfies the given epsilon value.

Are all continuous functions proven using the epsilon-delta proof?

No, not all continuous functions are proven using the epsilon-delta proof. This method is typically used for more complex functions or in situations where the continuity of a function needs to be rigorously proven, such as in mathematical analysis or theoretical physics. In simpler cases, the continuity of a function can be observed visually or proven using other methods.

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