Question

Is the function \( f(x) \) defined by

\[ f(x) = \begin{cases} x + 5, & \text{if } x \leq 1 \\ x - 5, & \text{if } x > 1 \end{cases} \]

continuous at \( x = 1 \)?

Hint:

To check for continuity at a point, ensure that the left-hand and right-hand limits as \( x \) approaches the point are equal to the function's value at that point.

Answer 1

Step 1: To determine continuity at \( x = 1 \), we check if the left-hand limit, right-hand limit, and the function value at \( x = 1 \) are equal. 

Step 2: The left-hand limit as \( x \to 1^- \) is: \[ \lim_{x \to 1^-} f(x) = 1 + 5 = 6 \] 

Step 3: The right-hand limit as \( x \to 1^+ \) is: \[ \lim_{x \to 1^+} f(x) = 1 - 5 = -4 \] 

Step 4: Since the left-hand limit and the right-hand limit are not equal, the function is not continuous at \( x = 1 \).