Determine the points of discontinuity for the function $y = u^2$, where $u = \begin{cases} x-1 & e

Question

Determine the points of discontinuity for the function $y = u^2$, where $u = \begin{cases} x-1 & 	ext{for } x \ge 0 \ x+1 & 	ext{for } x < 0 \end{cases}$.

Professor PC Thomas Classes · Answer

The function $u(x)$ is discontinuous at $x=0$. We check the composite function $y(x)$ at $x=0$. The right-hand limit is $\lim_{x 	o 0^+} y = \lim_{x 	o 0^+} (x-1)^2 = 1$. The left-hand limit is $\lim_{x 	o 0^-} y = \lim_{x 	o 0^-} (x+1)^2 = 1$. The function value is $y(0) = (0-1)^2 = 1$. Since $\lim_{x 	o 0} y(x) = y(0)$, the function is continuous at $x=0$ and thus continuous everywhere.