Math Problem Statement

Fungsi f(x) dikatakan kontinu di titik x = a jika: A. f(a) = lim x → a+ f(x) B. f(a) = lim x → a− f(x) C. f(a) = 0 D. lim x → a f(x) = f(a) E. lim x → a f(x) = 0

Solution

The question is asking under what conditions the function f(x)f(x) is continuous at the point x=ax = a. The options present different mathematical statements, and we need to determine which one correctly describes the condition for continuity.

For a function to be continuous at a point x=ax = a, three conditions must be satisfied:

  1. The function is defined at aa, meaning f(a)f(a) exists.
  2. The limit as xx approaches aa of the function exists, meaning limxaf(x)\lim_{x \to a} f(x) exists.
  3. The function value and the limit must be equal, meaning limxaf(x)=f(a)\lim_{x \to a} f(x) = f(a).

Now, let's analyze the options:

  • A. f(a)=limxa+f(x)f(a) = \lim_{x \to a^+} f(x): This is only the right-hand limit and does not include the left-hand limit. This is not enough to ensure continuity.

  • B. f(a)=limxaf(x)f(a) = \lim_{x \to a^-} f(x): This is only the left-hand limit and similarly does not account for the right-hand limit.

  • C. f(a)=0f(a) = 0: This does not address continuity in general, but only defines a specific function value at aa, which is irrelevant for continuity.

  • D. limxaf(x)=f(a)\lim_{x \to a} f(x) = f(a): This statement is the formal definition of continuity at a point because it includes the fact that the limit as xax \to a from both sides equals the function value at aa.

  • E. limxaf(x)=0\lim_{x \to a} f(x) = 0: This would only state that the limit of the function is zero at aa, which is unrelated to the general condition for continuity.

Thus, the correct answer is D.

Would you like more details on the concept of continuity, or do you have further questions? Here are some related questions:

  1. What are left-hand and right-hand limits in continuity?
  2. How do we check the continuity of a function graphically?
  3. Can a function be continuous at some points and not others?
  4. What is the difference between continuity and differentiability?
  5. How does continuity relate to limits in higher dimensions?

Tip: Always remember to check both the function value and the two-sided limit when verifying continuity.

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Math Problem Analysis

Mathematical Concepts

Continuity of Functions
Limits

Formulas

lim x → a f(x) = f(a)
lim x → a+ f(x)
lim x → a− f(x)

Theorems

Continuity at a Point

Suitable Grade Level

Grades 11-12 (or early college calculus)