# Uniform Continuity: Int x to x^2 e^(-t^2) on (1,∞)

• springo
In summary, the conversation discussed the method of proving a function is uniformly continuous by showing it has a bounded derivative. The example given involved using this method to prove the uniform continuity of a function on a given interval. The second question asked whether proving a function has a bounded derivative is enough to satisfy the grader. The answer to this question is that it depends on the requirements of the grader.
springo

## Homework Statement

Is the following uniformly continuous on (1,∞)?
$$\int_{x}^{x^2}\: e^{-t^2}\mathrm{d}t$$

## The Attempt at a Solution

Quite honestly I don't know where to start. I mean, I'm positive I have to use the theorem that says that if for all ε > 0 there exists δ > 0 so that for all x and y, |x - y| < δ implies |f(x) - f(y)| < ε. But that's all I got...

If you can show a function has bounded derivative over it's domain, then you've shown it's uniformly continuous. If you don't have that theorem yet, just remember the definition of differentiable. It has the epsilons and deltas you need.

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$$\frac{\mathrm{d}}{\mathrm{d}x}\left(\int_{x}^{x^2}f(t)\,\mathrm{d}t \right)=2x\cdot e^{-x^4}-e^{-x^2}$$

Since x→2x, x→e-x4 and x→e-x2 are bounded on (1,∞), $$2x\cdot e^{-x^4}-e^{-x^2}$$ is bounded so it's proved.

Is all OK?
Thanks.

PS: I have a similar problem where you have to say whether this is true or false:
|f'(x)|<= K (constant) for -∞< x < ∞
0 < f(x) <= x2 for x >= 1
Then g(x) = f(x)/x is uniformly continuous on (1,∞).
xf'(x)/x2 is bounded and f(x)/x2 too therefore true.

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If you've proved everything is bounded (like x*e^(-x^4), which does have to be proved, since it's an infinity*0 type of limit - 2x isn't bounded) and you have a theorem that says 'bounded derivative' -> 'uniformly continuous' and don't have to do an epsilon/delta proof, yes.

Oops, yeah I meant 2x is bounded if you don't consider infinity. It's easy to prove that limit, just set u = x4 and have u1/4/eu. What about the other question?
Thanks.

The other question was just if that proving it had a bounded derivative is enough to satisfy your homework grader. Only you can answer that.

Thanks a lot... I think I'm ready for tomorrow's exam now (lol)

## 1. What is uniform continuity?

Uniform continuity is a mathematical concept that describes a function's behavior over a given interval. A function is considered uniformly continuous if the change in its output values is small when the input values are close together. In other words, as the input values approach each other, the output values do not vary significantly.

## 2. How is uniform continuity different from continuity?

Continuity and uniform continuity are related concepts, but they differ in their definitions and properties. A function is continuous if it has no breaks or jumps in its graph, while a function is uniformly continuous if it has no sudden changes in output values as the input values get closer to each other. In other words, uniform continuity is a stricter condition than continuity.

## 3. Is the function x^2 e^(-t^2) uniformly continuous on the interval (1,∞)?

Yes, the function x^2 e^(-t^2) is uniformly continuous on the interval (1,∞). This can be proven using the epsilon-delta definition of uniform continuity, which states that for any given epsilon (ε) value, there exists a delta (δ) value such that if the distance between two input values is less than δ, then the difference between the corresponding output values is less than ε.

## 4. What is the importance of uniform continuity in mathematics?

Uniform continuity is an essential concept in mathematics because it helps us understand the behavior of functions and their graphs. It also allows us to prove the convergence of certain sequences and series, which is crucial in many areas of mathematics, such as calculus, analysis, and differential equations.

## 5. Can a function be uniformly continuous on one interval but not on another?

Yes, a function can be uniformly continuous on one interval but not on another. For example, the function x^2 e^(-t^2) is uniformly continuous on the interval (1,∞), but it is not uniformly continuous on the interval (0,1). This is because the function has a singularity at t = 0, which causes sudden changes in output values as the input values get closer to each other in this interval.

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