Question

The function $f$ defined by \[ f(x) = \begin{cases} x, & \text{if } x \leq 1 \\ 5, & \text{if } x>1 \end{cases} \] is not continuous at :

• A. $x = 0$
• B. $x = 1$
• C. $x = 2$
• D. $x = 5$

Hint:

To check for continuity at a point, ensure that the left-hand limit, right-hand limit, and the function value at that point are all equal.

Answer 1

The Correct Option is B

To check continuity at $x = 1$, we need to verify that the left-hand and right-hand limits at $x = 1$ are equal to the function value at $x = 1$. - The left-hand limit at $x = 1$ is $f(1) = 1$. - The right-hand limit at $x = 1$ is $f(1^+) = 5$. Since the left-hand and right-hand limits are not equal, the function is not continuous at $x = 1$.