This question aims to find **the set of points** at which the function is continuous if the points **( x, y )** of the given function are not equal to **( 0, 0 )**.

A **function** is defined as the **expression** which gives an output of the given input such that if we put **values of** **x** in the equation, it will give exactly **one value of y**. For example:

\[ y = x ^ 4 + 1 \]

This expression can be written in the form of function as:

\[ f ( y ) = x ^ 4 + 1 \]

## Expert Answer

The given function is $ f ( x, y) = \frac { x ^ 2 y ^ 3 } { 2 x ^ 2 + y ^ 2} $. The function f ( x ) is a **rational function** and every point in its** domain** makes it a continuous function. We have to check the continuity of function **f ( x, y )** at the origin. We will limit the function as:

\[ Lim _ { ( x, y ) \implies ( 0, 0 ) } f ( x, y ) = f ( 0, 0 ) \]

We have to check along the line by putting the value of **y = 0** in the function:

\[ Lim _ { x \implies 0 } = \frac { x ^ 2 ( 0 ) ^ 3 } { 2 x ^ 2 + ( 0 ) ^ 2 }\]

\[ Lim _ { x \implies 0 } = 0 \]

This means that the function **f ( x, y )** must be zero when its limit is such that ( x, y ) equals ( 0, 0 ). The value of **f ( 0, 0 )**

does not satisfy this condition. Hence, a function is said to be **continuous** if the **set of points** makes it continuous at the **origin**.

## Numerical Results

**The given function $ f ( x, y) \frac { x ^ 2 y ^ 3 } { 2 x ^ 2 + y ^ 2} $ is not a continuous function.**

## Example

Determine the** set of points** at which the **function** is **continuous** when the function is given as:

\[ f ( x , y ) = \frac { y ^ 2 x ^ 3 } { 3 y ^ 3 + ( y ) ^ 2 } \]

We have to check the continuity of function f ( x ) at the origin. We will limit the function as:

\[ Lim _ { ( x, y ) \implies ( 0, 0 ) } f ( x, y ) = f ( 0, 0 ) \]

\[ Lim _ { x \implies 0 } = \frac { y ^ 2 x ^ 3 } { 3 y ^ 3 + y ^ 2 } \]

We have to check along the line by putting the value of **y = 0** in the function:

\[ f ( 0, 0) = \frac { 0^ 2 x ^ 3 } { 3 (0) ^ 3 + ( 0 ) ^ 2 } \]

\[ Lim _ { x \implies 0 } = 0 \]

This means that the function f ( x, y ) must be zero when its limit is such that ( x, y ) equals ( 0, 0 ). The value of f ( 0, 0 ) does not satisfy this condition. **The given function is not continuous at the origin**.

*Image/Mathematical drawings are created in Geogebra**.*