# Proving Continuity of f(x)=x^2sin(pi/x) at x=0

• zeroheero
In summary, the conversation discusses using the squeeze theorem to prove the continuity of f(x) = x^2sin(pi/x) at x = 0. The squeeze theorem is applied by adding x^2 to the inequality and using the functions g(x) = -x^2, f(x) = x^2sin(pi/x), and h(x) = x^2. This is considered the most efficient method for solving the assignment question.
zeroheero
Hi, I have an assignment question that asks if f(x) = x^2sin(pi/x), prove that f(0) can be defined in such a way the f becomes continuous at x = 0.
Am I able to apply the squeeze theorem to show,
-1<sin(pi/x)<1
-x\<x^2sin(pi/x)\<x^2. (\< us less than or equal to)
Lim as x approaches 0 from the left side, -x^2=0; and
Lim as x approaches 0 from the right side, x^2=0
if g(x) =-x^2 F(x) = x^2sin(pi/x). h(x)= x^2

Pretty much. It is the easiest way without invoking other theorems.

## 1. What is Continuity?

Continuity is a mathematical concept that describes a function's behavior at a specific point. A function is considered continuous if it has no sudden jumps or breaks at that point, and its graph can be drawn without lifting the pencil.

## 2. How do you prove the continuity of a function?

To prove the continuity of a function at a point, you need to show that the limit of the function at that point exists and is equal to the value of the function at that point. In other words, the left-hand limit, right-hand limit, and the value of the function at the point must be equal.

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

The function f(x)=x^2sin(pi/x) is a piecewise function that combines the quadratic function x^2 and the trigonometric function sin(pi/x). It is defined for all real numbers except x=0, where it is undefined due to the division by zero.

## 4. How do you prove the continuity of f(x)=x^2sin(pi/x) at x=0?

To prove the continuity of f(x)=x^2sin(pi/x) at x=0, we need to show that the limit of the function at x=0 exists and is equal to the value of the function at x=0. We can do this by evaluating the left-hand limit, right-hand limit, and the value of the function at x=0 and showing that they are all equal.

## 5. What is the significance of proving continuity at x=0 for f(x)=x^2sin(pi/x)?

Proving continuity at x=0 for f(x)=x^2sin(pi/x) is important as it establishes that the function has no sudden jumps or breaks at this point. This means that the function can be smoothly drawn without lifting the pencil, and it is considered a continuous function. It also allows us to apply various calculus techniques, such as finding derivatives and integrals, to the function at x=0.

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