Let $f ( x , y , z ) = \left\{ \begin{array} { c l }
\frac { x - y - z } { x ^ { 2 } - ( y + z ) ^ { 2 } } & x ^ { 2 } \neq ( y + z ) ^ { 2 } \
\frac { 1 } { 4 } & x ^ { 2 } = ( y + z ) ^ { 2 }
\end{array} \right.$ . Then f is

A)Continuous at (2,1,1)
B)Discontinuous at (2,1,1) because f is undefined at(2,1,1)
C)Discontinuous at (2,1,1) because the limit of f at (2,1,1) does not exist
D)Discontinuous at (2,1,1) because the limit of f at (2,1,1) is different from the function values at (2,1,1)
E)Discontinuous at (2,1,1) for a reason that is different from the ones above

Question

Let $f ( x , y , z ) = \left\{ \begin{array} { c l } 
\frac { x - y - z } { x ^ { 2 } - ( y + z ) ^ { 2 } } & x ^ { 2 } \neq ( y + z ) ^ { 2 } \
\frac { 1 } { 4 } & x ^ { 2 } = ( y + z ) ^ { 2 }
\end{array} \right.$ . Then f is

A)Continuous at (2,1,1)
B)Discontinuous at (2,1,1) because f is undefined at(2,1,1)
C)Discontinuous at (2,1,1) because the limit of f at (2,1,1) does not exist
D)Discontinuous at (2,1,1) because the limit of f at (2,1,1) is different from the function values at (2,1,1)
E)Discontinuous at (2,1,1) for a reason that is different from the ones above

Quizplus · Accepted Answer

DAt the point (2,1,1), $x^2 = (y+z)^2$ holds true since $2^2 = (1+1)^2 = 4$. According to the definition of $f$, when $x^2 = (y+z)^2$, $f(x,y,z) = \frac{1}{4}$. Therefore, the function is defined at (2,1,1) and equals $\frac{1}{4}$. To check for continuity, we need to see if the limit of $f$ as $(x,y,z)$ approaches (2,1,1) equals the function value at (2,1,1). The limit of the function as it approaches (2,1,1) from any path is $\frac{1}{4}$, matching the function's value at that point, indicating continuity. However, the function's definition outside the condition $x^2 = (y+z)^2$ suggests a potential for discontinuity if not for the specific handling of the case where $x^2 = (y+z)^2$. Thus, the function is continuous at (2,1,1), but the reasoning involves understanding both the specific value at (2,1,1) and the behavior of the function around that point.

Let Then F Is A)Continuous at (2,1,1) B)Discontinuous at \left

Let Then F Is

A)Continuous at (2,1,1)
B)Discontinuous at \left