Question

Let the function $ f(x) $ be defined as $ f(x) = \frac{x - |x|}{x} $, then:

• A. the function is continuous everywhere
• B. the function is not continuous
• C. the function is continuous when $ x < 0 $
• D. the function is continuous for all $ x $ except zero

Hint:

Always check the domain first. If a function's denominator is zero at a point, it cannot be continuous there, regardless of whether the limit exists.

Answer 1

The Correct Option is D

Concept: A function is continuous where it is defined and the limit equals the function value. First, we must redefine $ f(x) $ by removing the absolute value sign based on the domain of $ x $.

Step 1: Redefining the function for different intervals. Recall the definition of $ |x| $: $$ |x| = \begin{cases} x & \text{if } x > 0
-x & \text{if } x < 0 \end{cases} $$ Note that at $ x = 0 $, the denominator of $ f(x) $ is zero, so the function is undefined at $ x = 0 $.
• For $ x > 0 $: $ f(x) = \frac{x - x}{x} = \frac{0}{x} = 0 $.
• For $ x < 0 $: $ f(x) = \frac{x - (-x)}{x} = \frac{x + x}{x} = \frac{2x}{x} = 2 $.

Step 2: Analyzing continuity.
The function is a constant $ 0 $ for all $ x \in (0, \infty) $, which is continuous. The function is a constant $ 2 $ for all $ x \in (-\infty, 0) $, which is also continuous. However, at $ x = 0 $, the function is not defined. Furthermore, the left-hand limit ($ 2 $) is not equal to the right-hand limit ($ 0 $), showing a jump discontinuity.

Step 3: Conclusion.
Since the function is defined and constant in the intervals $ (-\infty, 0) $ and $ (0, \infty) $, it is continuous on those intervals. The only point of concern is $ x=0 $, where the function fails to exist. Thus, the function is continuous for all $ x $ except zero.