Continuous Extension


Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.

Verified step by step guidance

1. To understand why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0, we first need to consider the behavior of the function as 𝓍 approaches 0. The function is defined for all 𝓍 ≠ 0, but we are interested in its behavior as 𝓍 gets very close to 0.
2. As 𝓍 approaches 0, the expression 1/𝓍 becomes very large in magnitude, which means that the argument of the sine function oscillates rapidly between positive and negative values. This rapid oscillation causes the function ƒ(𝓍) = sin(1/𝓍) to oscillate between -1 and 1 without settling down to any particular value.
3. For a function to have a continuous extension at a point, the limit of the function as it approaches that point must exist and be finite. In this case, we need to check if the limit of ƒ(𝓍) as 𝓍 approaches 0 exists.
4. To determine the limit, consider the fact that for any sequence of values of 𝓍 approaching 0, the corresponding sequence of values of 1/𝓍 will cover all real numbers densely. This means that the values of sin(1/𝓍) will cover the interval [-1, 1] densely as well, without converging to a single value.
5. Since the limit of ƒ(𝓍) as 𝓍 approaches 0 does not exist (the function does not approach a single value), there is no way to define ƒ(0) such that the function becomes continuous at 𝓍 = 0. Therefore, ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. For the function ƒ(𝓍) = sin(1/𝓍), we need to analyze the limit as 𝓍 approaches 0. If the limit does not exist or is not finite, the function cannot be continuously extended to that point.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For ƒ(𝓍) = sin(1/𝓍}, we find that as 𝓍 approaches 0, the function oscillates between -1 and 1, indicating that it does not settle at a single value, thus failing the continuity requirement at 𝓍 = 0.

Oscillation

Oscillation refers to the behavior of a function that fluctuates between values without converging to a single limit. In the case of ƒ(𝓍) = sin(1/𝓍), as 𝓍 approaches 0, the function oscillates infinitely between -1 and 1, which means it does not approach any specific value, preventing a continuous extension to 𝓍 = 0.