Chapter 1.5, Problem 34ES

A function $f$ is said to have a removable discontinuity at $x=c$ if ${\lim }_{x\to c}f\left(x\right)$ exists but $f$ is not continuous at $x=c,$ either because $f$ is not defined at $c$ or because the definition for $f\left(c\right)$ differs from the value of the limit. This terminology will be needed in these exercises.

(a) The terminology removable discontinuity is appropriate because a removable discontinuity of a function $f$ at $x=c$ can be “removed� by redefining the value of $f$ appropriately at $x=c$ . What value for $f\left(c\right)$ removes the discontinuity?

(b) Show that the following functions have removable discontinuities at $x=1,$ and sketch their graphs.

$f\left(x\right)=\frac{x^2-1}{x-1}$ and $g\left(x\right)=\left\{\begin{array}{ll}1,\hfill & x>1\hfill \\ 0,\hfill & x=1\hfill \\ 1,\hfill & x<1\hfill \end{array}\right.$

(c) What values should be assigned to $f\left(1\right)$ and $g\left(1\right)$ to remove the discontinuities?