Continuity Of Functions: How To Determine Existence Of F(2)

Which of the following statements, if true, can be used to conclude that f(2) exists?i. limx→2f(x) exists.ii. f is continuous at x=2.iii. f is differentiable at x=2.

II and III onlyIf ff is continuous at x=2, then f(2) exists. Also if f is differentiable at x=2, then ff is continuous at x=2 and f(2) exists.

Statement ii. f is continuous at x = 2 can be used to conclude that f(2) exists.

Here’s why:

Continuity of a function at a point means that the function exists, the limit of the function at that point exists, and the limit equals the function value at that point.

So, if f is continuous at x = 2, it means that f(2) exists and is equal to the limit of the function as x approaches 2. This is different from just saying that the limit exists, as in statement i. The limit existing only tells us that the function approaches a certain value as x gets closer to 2, but doesn’t necessarily tell us what the function value is at x = 2.

Statement iii. f is differentiable at x = 2 is also related to the existence of f(2), but it doesn’t directly imply that it exists. Differentiability means that the function has a derivative (slope) at that point, which is a stronger condition than continuity.

In summary, if f is continuous at x = 2, then we can conclude that f(2) exists.

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