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 determine if a function has a continuous extension at a point, we need to check if the limit of the function exists as it approaches that point. In this case, we are examining the limit of f(x) = sin(1/x) as x approaches 0.
2. Consider the behavior of the function f(x) = sin(1/x) as x approaches 0. As x gets closer to 0, 1/x becomes very large, causing the argument of the sine function to oscillate rapidly between positive and negative values.
3. The sine function, sin(θ), oscillates between -1 and 1 for any real number θ. Therefore, as x approaches 0, sin(1/x) does not settle towards any particular value but continues to oscillate between -1 and 1.
4. For a limit to exist at a point, the function must approach a single, finite value as x approaches that point. However, due to the rapid oscillation of sin(1/x) as x approaches 0, the limit does not exist.
5. Since the limit of f(x) = sin(1/x) as x approaches 0 does not exist, the function cannot be continuously extended to x = 0. A continuous extension would require the function to have a well-defined limit at that point.