High School

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In order for a function f(x) to be continuous at an interior point x=a of its domain, what condition should hold true?

a. lim f(x) = f(a)

b. lim f(x-a) = lim f(x+a) = f(x), as a→0

c. lim f(x-h) = lim f(x+h) = f(x), as h→a = lim f(a - h)

d. lim f(a + h) = f(a), as h→0

Answer :

To determine the condition under which a function [tex]f(x)[/tex] is continuous at an interior point [tex]x = a[/tex] of its domain, we need to understand the definition of continuity in calculus.

For a function [tex]f(x)[/tex] to be continuous at [tex]x = a[/tex], the following condition must hold:


  1. The function [tex]f(x)[/tex] is defined at [tex]x = a[/tex]. This means that [tex]f(a)[/tex] exists.


  2. The limit of [tex]f(x)[/tex] as [tex]x[/tex] approaches [tex]a[/tex] should exist. This means that both [tex]\lim_{x \to a^-} f(x)[/tex] (the limit from the left) and [tex]\lim_{x \to a^+} f(x)[/tex] (the limit from the right) must exist and be equal.


  3. The value of the limit should be equal to the value of the function at that point. Therefore, [tex]\lim_{x \to a} f(x) = f(a)[/tex].



Based on these criteria, option a. [tex]\lim_{x \to a} f(x) = f(a)[/tex] is the correct condition for continuity at the point [tex]x = a[/tex].

Thus, for a function to be continuous at [tex]x = a[/tex], the function must be defined at [tex]a[/tex], the limit as [tex]x \to a[/tex] must exist, and this limit must equal the function's value at [tex]a[/tex].

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