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

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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