Q. 2 s - Calculus

Question

Examples: 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

(a). The example of a function$f\left(x\right)=\frac{x^2-6x+8}{x-4}$.

which has a removable discontinuity at x=4.

(b). The example of the function that is continuous in its domain but not continuous at x=0 is$f\left(x\right)=\ln x$.

(c). The example of the function that is continuous on (0,2] and (2,3) but not continuous on (0,3) is,

f(x)= \begin{cases}x-5  \ if  0<x \leq 2 \\ x^{2}  \  if  2<x<3\end{cases}

Part (a) Step 1: Given information

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

Part (a) Step 2: Simplification

Consider the following function,

$f\left(x\right)=\frac{(x-2)(x-4)}{x-4}$

It can be seen that the value of the function is 2 at x=4 and it has a removable discontinuity at the point x=4.

The function can be rewritten as,

$f\left(x\right)=\frac{x^2-6x+8}{x-4}$

Therefore, the example of a function $f\left(x\right)=\frac{x^2-6x+8}{x-4}$which has a removable discontinuity at x=4.

Part (b) Step 1: Given information

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

Part (b) Step 2: Simplification

Consider the natural logarithm function,

$f\left(x\right)=\ln x$

The domain of the function is $x\in (0,\infty ).$

The natural logarithm function is continuous in its domain and it is not defined at x=0.

Therefore, the example of the function that is continuous in its domain but not continuous at x=0 is $f\left(x\right)=\ln x.$

Part (c) Step 1: Given information

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

Part (c) Step 2: Simplification

Consider the following piecewise defined function,

f(x)= \begin{cases}x-5 & \text { if } 0<x \leq 2 \\ x^{2} & \text { if } 2<x<3\end{cases}

It can be seen that above function is continuous in the interval (0,2) and (2,3).

From the first definition, the value of the function at the point x=2 is,

$$\begin{gathered} f\left(2\right)=2-5 \\ =-3 \end{gathered}$$

From the second definition, the value of the function at the point $x=2$is,

$$\begin{gathered} f\left(2\right)=2^2 \\ =4 \end{gathered}$$

It can be seen that the function is not continuous in the interval (0,3) as there is a discontinuity at the point x=2.

Therefore, the example of the function that is continuous on (0,2] and (2,3) but not continuous on (0,3) is,

f(x)= \begin{cases}x-5 & \text { if } 0<x \leq 2 \\ x^{2} & \text { if } 2<x<3\end{cases}