Question

Find the function in which the function having a jump discontinuity among the following.

• A. $f\left(x\right)=\{\begin{array}{c}{x}^2+1\forall x\in R-\left\{10\right\}\\ 15x=10\end{array}$
• B. $f\left(x\right)=\{\begin{array}{c}{x}^2+1\forall x\in (-\infty ,5)\\ x^2-1\forall x\in [5,\infty ]\end{array}$
• C. $f\left(x\right)=\frac{x}{x-1}$
• D. $f\left(x\right)=\frac{x}{x^2-1}$

Answer 1

The correct option is B

$f\left(x\right)=\{\begin{array}{c}{x}^2+1\forall x\in (-\infty ,5)\\ x^2-1\forall x\in [5,\infty ]\end{array}$

Here, for the function to have jump discontinuity.
The limit from left and right should be finite and unequal. We can either evaluate the limits for such special points or we can evaluate graphically.

Let's also see the other options

Option (a.)

$f\left(x\right)=\{\begin{array}{c}{x}^2+1x\in R-10\\ 15,x=10\end{array}$

Here the discontinuity is of the removable type since the discontinuity is removable by redefining the function's value at $x=10$.

Option (c.) and (d.) $f\left(x\right)=\frac{1}{x-1},$
Here we have limit of infinity, same is true for
$f\left(x\right)=\frac{1}{x^2-1}$

Option (b.)

f(x) = $\{\begin{array}{c}{x}^2+1\forall x\in (-\infty ,5)\\ x^2-1\forall x\in [5,\infty ]\end{array}$

Its clear from the figure that the discontinuity is of jump type.