Q. 2 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading:

(a) The graph of a function with $f\left(4\right)=2$ that has a removable discontinuity at $x=4.$

(b) The graph of a function that is continuous on its domain but not continuous at $x=0.$

(c) The graph of a function that is continuous on $(0,2] and (2,3)$ but not on $(0,3).$

Step-by-Step Solution

Answer

Part (a) The graph of a function with $f\left(4\right)=2$ that has a removable discontinuity at $x=4.$

Part (b) The graph of a function that is continuous on its domain but not continuous at$x=0.$

Part (c) The graph of a function which is continuous on $(0,2] and (2,3)$ but not continuous on $(0,3).$

Part (a) Step 1. Calculation

The function $f\left(4\right)=2$ that has a removable discontinuity at $x=4$ is $f\left(x\right)=\left\{\begin{array}{l}\frac{x}{2}:0<x\le 4\\ 5x:4<x<5\end{array}\right.$

Here,

$LHL\ne RHL at x=4$

The graph is given below

Part (b) Step 1. Calculation

A function that is continuous on its domain but not continuous at $x=0$

$$\begin{gathered} f\left(x\right)=1/x \\ {\left(Domain\right)}_{f\left(x\right)}=R-\{0 \end{gathered}$$

The graph is given below

Part (c) Step 1. Calculation

A function which is continuous on $(0,2] and (2,3)$ but not continuous on $(0,3)$ is

$f\left(x\right)\left\{\begin{array}{l}x+2:0<x\le 2\\ 4+x:2<x<3\end{array}\right.$

Here function is continuous on $(0,2] and (2,3)$ but not continuous on $(0,3)$.

The graph is given below

Here function is continuous on $(0,2] and (2,3)$ but not continuous on $(0,3).$