Question

If a function $ f(x) $ defined on $ [a, b] $ is discontinuous at $ x = \alpha \in (a, b) $, then:

• A. \( \lim_{x \to \alpha^-} f(x) = \lim_{x \to \alpha^+} f(x) = f(\alpha) \)
• B. \( \lim_{x \to \alpha^-} f(x) = f(\alpha) \)
• C. \( \lim_{x \to \alpha^-} f(x) = f(a) \)
• D. \( \lim_{x \to \alpha^-} f(x) = f(b) \)

Hint:

A function can be discontinuous due to a jump (\( \lim_{x \to \alpha^-} \neq \lim_{x \to \alpha^+} \)), a removable discontinuity (\( \lim_{x \to \alpha} \neq f(\alpha) \)), or an infinite discontinuity.

Answer 1

The Correct Option is B

Step 1: Understand the discontinuity at \( x = \alpha \).
A function is continuous at \( x = \alpha \) if \( \lim_{x \to \alpha^-} f(x) = \lim_{x \to \alpha^+} f(x) = f(\alpha) \). Discontinuity means this fails: either the limits differ, the limit doesn’t equal \( f(\alpha) \), or the limit doesn’t exist.
Step 2: Analyze the options.
Option 1: Implies continuity, which contradicts the problem.
Option 2: The left-hand limit equals \( f(\alpha) \), but if the right-hand limit differs, the function is discontinuous (e.g., jump discontinuity).
Options 3 and 4: Unrelated to discontinuity at \( \alpha \), as they involve \( f(a) \) and \( f(b) \).
Step 3: Conclusion.
Option 2 allows for discontinuity (e.g., if \( \lim_{x \to \alpha^+} f(x) \neq f(\alpha) \)), making it the correct choice.