Animated Solution for Mathematics - Limits, Continuity and Differentiability: If $x+|y|=2y$, then $y$ as a function of $x$ is

The equation provided is $x+|y|=2y$. The presence of the absolute value $|y|$ indicates that the behavior of the function depends on the sign of $y$.

To solve this, we must analyze the definition of the modulus: $|y|=y$ if $y\ge 0$ $|y|=-y$ if $y<0$

In the first universe, we assume $y\ge 0$. The equation simplifies to:

Subtracting $y$ from both sides, we obtain $y=x$. Given our constraint $y\ge 0$, this branch is valid only for $x\ge 0$.

In the second universe, we assume $y<0$. Here, the modulus acts as a negation, so $|y|=-y$. The equation transforms into:

Rearranging this, we find $x=3y$, or:

Since we assumed $y<0$, it follows that $\frac{x}{3}<0$, which implies $x<0$.

We have constructed a piecewise function:

This function is defined for all real $x$. To check for continuity at $x=0$, we evaluate the limits:

Since the left-hand limit, the right-hand limit, and the function value $f(0)=0$ are all equal, the graph is continuous at the origin.

Differentiability measures the smoothness of the function. For $x<0$, the slope is the derivative of $\frac{x}{3}$, which is $\frac{1}{3}$.

At $x=0$, we compare the one-sided derivatives: The left-hand derivative is $\frac{1}{3}$. The right-hand derivative is the derivative of $x$, which is $1$.

Because $\frac{1}{3} eq 1$, the graph possesses a sharp "kink" at the origin. Consequently, the function is not differentiable at $x=0$.