Q. 46 - Calculus

Question

Use graphs to determine if each function is continuous at the given point x = c. 

$f\left(x\right)=\left\{\begin{array}{l}{x}^2-3,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ rational }\\ 3x+1,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ irrational, }\end{array}c=0\right.$

Step-by-Step Solution

Answer

The function is not continuous at x = 0.

Step 1. Given information.

We have been given a function as:

$f\left(x\right)=\left\{\begin{array}{l}{x}^2-3,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ rational }\\ 3x+1,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ irrational, }\end{array}c=0\right.$

We have to determine if this function is continuous at the given point x = c. 

Step 2. Draw the graph.

Step 3. Analyze the graph for the continuity of the function at x = c.

From the graph, we can see that:

For the rational values of x, $\underset{x\to 0}{lim} f\left(x\right)=-3$

For the irrational values of x, $\underset{x\to 0}{lim} f\left(x\right)=1$

These two values are different.

Therefore, this function is not continuous at x = 0.