Limits and Continuity


On what intervals are the following functions continuous?


b. g(x) = csc x

Verified step by step guidance

1. Identify the function: The function given is \( g(x) = \csc x \), which is the cosecant function. Recall that \( \csc x = \frac{1}{\sin x} \).
2. Determine where the function is undefined: Since \( \csc x \) is the reciprocal of \( \sin x \), it is undefined wherever \( \sin x = 0 \).
3. Find the zeros of \( \sin x \): The sine function is zero at integer multiples of \( \pi \), i.e., \( x = n\pi \) where \( n \) is an integer.
4. Identify intervals of continuity: The function \( g(x) = \csc x \) is continuous on intervals where \( \sin x \neq 0 \). These intervals are between the points where \( \sin x = 0 \), i.e., between \( n\pi \) and \( (n+1)\pi \) for any integer \( n \).
5. Conclude the intervals: Therefore, \( g(x) = \csc x \) is continuous on the intervals \( (n\pi, (n+1)\pi) \) for all integers \( n \).

Key Concepts

Continuity of Functions

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 a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is crucial for determining where functions do not have breaks, jumps, or asymptotes.

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is important to note that csc(x) is undefined wherever sin(x) = 0, which occurs at integer multiples of π. Understanding the behavior of the cosecant function helps identify the intervals of continuity.

Intervals of Continuity

Intervals of continuity refer to the ranges of x-values where a function is continuous. For the function g(x) = csc(x), we need to exclude points where the function is undefined, specifically at x = nπ (where n is an integer). By identifying these points, we can determine the intervals where g(x) is continuous, which are the open intervals between these undefined points.