# Show that f(x) = sin(x^2) is continuous for all a [-sqrt(pi), sqrt(pi).

• Robultronic
In summary, the conversation discusses the continuity of a function f on the interval [-√π,√π], where f(x)=sin(x^2). The goal is to show that f is continuous at every point in the interval and to find a value of delta that satisfies the given conditions for continuity. The solution involves using the definition of continuity and showing that for every point a in the interval, the limit of f(x) as x approaches a is equal to f(a). The conversation also touches on using other facts to simplify the proof.
Robultronic

## Homework Statement

Given f : [-$$\sqrt{\pi}$$, $$\sqrt{\pi}$$ ] $$\rightarrow$$ [-1, 1]

f(x) = sin(x$$^{2}$$)

a)
Show that f is continuous for all a $$\in$$ [-$$\sqrt{\pi}$$, $$\sqrt{\pi}$$ ]

b)
Find a $$\delta$$ so that |x - y| $$\leq$$ $$\delta$$ implies that
|f(x) - f(y)| $$\leq$$ 0.1 for all x and y in [-$$\sqrt{\pi}$$, $$\sqrt{\pi}$$ ]

## The Attempt at a Solution

I honestly don't know where to begin. But would be very pleased for any help. A full solution would be the optimal though. But anything helps.

If you want to go back to the definition of continuity, you need to show that for every point a in the interval,

$$\lim_{x \to a} f(x) = f(a)$$

(with the appropriate one-sided limit at the endpoints). If you can assume other facts, then you can argue f(x) is continuous in a less tedious way. What do you think you're supposed to do?

I'm supposed to use the definition of continuity.

I know how to show continuity at a point, but have no clue of how to do it for an interval.

You just have to show f is continuous at every point in the interval. It essentially amounts to saying "Let a∈[-√π,√π]" and showing f is continuous at x=a like usual.

Lol, that simple, thx. I kept thinking I had to use the interval in the proof somehow and not just say a = the interval. OK, that I can do. How about b?

I think I solved it, I will write it down when I get home so someone can tell if its correct or not, but have to go now.

## 1. What is the definition of continuity?

The definition of continuity is the property of a function where small changes in the input result in small changes in the output. In other words, a function is continuous if it has no abrupt changes or breaks in its graph.

## 2. How do you prove that a function is continuous?

In order to prove that a function is continuous, you must show that it satisfies the three conditions of continuity: the function is defined at the point in question, the limit of the function exists at that point, and the limit is equal to the value of the function at that point.

## 3. What is the domain of the function f(x) = sin(x^2)?

The domain of the function f(x) = sin(x^2) is all real numbers, or (-∞, ∞). This means that the function is defined for all values of x.

## 4. How do you show that f(x) = sin(x^2) is continuous for all a [-sqrt(pi), sqrt(pi)?

In order to show that f(x) = sin(x^2) is continuous for all a [-sqrt(pi), sqrt(pi)], we must show that it satisfies the three conditions of continuity for all values of a within this interval. This can be done by evaluating the limit of the function at a, and then showing that it is equal to the value of the function at a.

## 5. Why is it important to prove that a function is continuous?

Proving that a function is continuous is important because it guarantees that the function will behave in a predictable manner, and that there will be no abrupt changes or breaks in its graph. This is essential for many applications in mathematics and science, where continuity is a fundamental property of functions.

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