Determine whether the function is continuous at the given...

Name: Problem 46, Chapter 1: Calculus
Uploaded: 2021-06-14T17:49:45-08:00
Duration: 1 min 7 s
Channel: Carson Merrill
Description: Determine whether the function is continuous at the given point c. If the function is not continuous, determine whether the discontinuity is removable or non removable. F(x)=x sin(1)/(x) ; c=0

Determine whether the function is continuous at the given point c. If the function is not continuous, determine whether the discontinuity is removable or non removable.
$$F(x)=x \sin \frac{1}{x} ; c=0$$

Key Concepts

Removable and Non-Removable Discontinuities

These terms describe types of interruptions in a function. A removable discontinuity occurs when the limit of the function exists at a point, but the function is either not defined there or is defined to take a different value than the limit. In contrast, a non-removable discontinuity occurs when the limit does not exist or the behavior of the function near the point is such that the discontinuity cannot be resolved simply by redefining the function at that point.

Squeeze Theorem

The Squeeze Theorem is a useful tool in calculus for evaluating limits. It states that if a function is 'squeezed' between two other functions that share the same limit at a particular point, then the original function must also converge to that same limit. This theorem is particularly helpful when dealing with functions that exhibit oscillatory behavior.

Continuity at a Point

This concept involves assessing whether a function at a specific point has no interruptions or jumps, meaning that the limit of the function as it approaches the point is equal to the function’s actual value at that point. It requires the function to be defined at the point, and the left-hand and right-hand limits must both exist and be equal to that value.

Limit of a Function

This is a fundamental concept that examines the behavior of a function as the variable approaches a particular value. In determining continuity at a specific point, the limit helps establish whether a small change in input near that point leads to predictable, stable changes in output, which is essential for understanding if a function 'flows' smoothly without abrupt changes.

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