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Determine the set of points at which the function is continuous.

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.

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