Q. 47 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The function is continuous at x = 1.

Step 1. Given information.

We have been given a function as:

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

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, $\underset{x\to 1}{lim} f\left(x\right)=1$

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

These two values are the same.

Therefore, this function is continuous at x = 1.