Prove continuity of sqrt(x) on (0,infinity)

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In summary, the conversation discusses proving the continuity of the function L(x) = √x on the interval (0,∞). The strategy for the proof is to let ε > 0 be given and assume |x - x_0| < δ, and then manipulate the expression for |L(x) - L(x_0)| to get a term in terms of (x - x_0) that can be controlled. One approach is to use inverse functions and set δ = ε^2. The conversation ends with a confirmation that this approach is valid.
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Homework Statement


This is a problem from my Analysis exam review sheet.

Let L(x) = [itex]\sqrt{x}[/itex]. Prove L is continuous on E = (0,[itex]\infty[/itex])

The Attempt at a Solution



The way we've been doing these proofs all semester is to let [itex]\epsilon > 0[/itex] be given, then assume [itex]\left| x -x_{0} \right| < \delta[/itex] (which we figure out later) and [itex] x_{0} \in E [/itex]

Then look at.
[itex]\left| L(x) - L(x_{0}) \right| = \left| \sqrt{x} - \sqrt{x_{0}}\right|[/itex]
and try to get a [itex]\left( x -x_{0} \right)[/itex] term which we can control, so that we can figure out a [itex]\delta[/itex] which will allow us to get the entire thing less than [itex]\epsilon[/itex]. So essentially I understand how to do the proof. My algebra skills are just really rusty and I can't figure out the long division or whatever I need to do to get a [itex]\left( x -x_{0} \right)[/itex] term out of that.
 
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  • #2
One trick is to abuse inverse functions whose continuity you can deal with

If you let [itex] y^2=x[/itex] and [itex] y_0^2 = x_0[/itex] then
[tex] |x-x_0| = |y^2-y_0^2| = |y_0-y||y_0+y|<\delta[/tex]

Now you want to show that [itex] |y-y_0|<\epsilon[/itex] if delta is small enough
 
  • #3
Thanks for the quick response, I think I see where to go from here, but just to make sure.
Let [itex]\delta = \epsilon^{2}[/itex]
We know [itex]\left| y + y_{0} \right| > \left| y - y_{0} \right|[/itex] since [itex]y[/itex] and [itex]y_{0}[/itex] are positive.
and since [itex]\left| y + y_{0} \right| \left| y - y_{0} \right| < \delta = \epsilon^{2}[/itex]
We can see that [itex]\left| y - y_{0} \right| < \sqrt{\delta} = \epsilon[/itex]
 

FAQ: Prove continuity of sqrt(x) on (0,infinity)

What is continuity?

Continuity is a mathematical concept that describes the smoothness and connectedness of a function. In simpler terms, a function is continuous if it does not have any abrupt changes or breaks in its graph.

How do you prove continuity of a function?

In order to prove continuity of a function, we must show that the function is defined at a certain point, the limit of the function exists at that point, and the value of the function at that point is equal to the limit. This can be done using the definition of continuity or using the three-part continuity theorem.

What is the definition of continuity?

The formal definition of continuity states that a function f(x) is continuous at a point a if and only if the limit of f(x) as x approaches a exists and is equal to f(a).

How do you prove continuity of the square root function on (0, infinity)?

To prove continuity of the square root function on the interval (0, infinity), we must show that the function is defined at every point on the interval, the limit of the function as x approaches a point on the interval exists, and the value of the function at that point is equal to the limit. This can be done using algebraic manipulation and the definition of continuity.

Why is proving continuity important in mathematics?

Proving continuity is important because it helps us understand the behavior of a function and its graph. It allows us to make accurate predictions and use the function in various mathematical applications. Additionally, continuity is a fundamental concept in calculus and is necessary for understanding more complex concepts such as limits, derivatives, and integrals.

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