Question

If $x+|y|=2y,$ then y as a function of x is

• A. defined for all real x
• B. continuous at x=0
• C. differentiable for all x
• D. such that dy/dx=1/3 for x< 0

Answer 1

Answer: (A), (B), (D)

such that dy/dx=1/3 for x< 0
continuous at x=0
defined for all real x

If $x+|y|=2y$, then $y$ can be written in terms of $x$ as

$y=\{\begin{array}{cc}x& \text{if}y\ge 0\\ \frac{1}{3}x& \text{if}y<0\end{array}$

Therefore, y as a function of x is defined for all x and is continuous at $x=0$,
but it is not differentiable at $x=0$, because

$f^{\prime }(0+)=\underset{h\to 0+}{lim}\frac{f(0+h)-f\left(0\right)}{h}=1$

and $f^{\prime }(0-)=\underset{h\to 0-}{lim}\frac{f(0+h)-f\left(0\right)}{h}=\frac{1}{3}$