Discuss the continuity of the functions f and g. Explain any differences

Lennie Carroll

Lennie Carroll

Answered question

2021-10-16

Discuss the continuity of the functions f and g. Explain any differences
f(x,y)={x2+2xy2+y2x2+y2,(x,y)00,(x,y)=0
g(x,y)={x2+2xy2+y2x2+y2,(x,y)01,(x,y)=0

Answer & Explanation

Viktor Wiley

Viktor Wiley

Skilled2021-10-17Added 84 answers

We will first consider the given functions,
The objective us to discuss the continuity of the functions.
First, we will consider lim(x,y)(0,0)x2+2xy2+y2x2+y2=lim(0,y)(0,0)0+2(0)+y20+y2
But f(0,0)=0
Now, as lim(x,y)(0,0)f(x,y)f(0,0)
Hence, we can say that the function f(x,y) is continuous everywhere except at (0,0).
Next, we will consider g(x,y)={x2+2xy2+y2x2+y2,(x,y)01,(x,y)=0
Now, we will find lim(x,y)(0,0)g(x,y) along the kine x=0, we get,
lim(x,y)(0,0)x2+2xy2+y2x2+y2=lim(0,y)(0,0)0+2(0)+y20+y2
=1
And g(0,0)=1
Now, as lim(x,y)(0,0)g(x,y)=g(0,0)
Hence, we can say that the function g(x,y) is continuous everywhere.

Jeffrey Jordon

Jeffrey Jordon

Expert2022-08-30Added 2605 answers

Answer is given below (on video)

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