Limits and Continuity


On what intervals are the following functions continuous?


b. g(x) = csc x

Verified step by step guidance

1. To determine the intervals where the function \( g(x) = \csc x \) is continuous, we first need to understand the definition of the cosecant function. The cosecant function is defined as \( \csc x = \frac{1}{\sin x} \).
2. The function \( \csc x \) will be continuous wherever \( \sin x \neq 0 \), because division by zero is undefined, leading to discontinuities.
3. The sine function, \( \sin x \), is zero at integer multiples of \( \pi \), i.e., \( x = n\pi \) where \( n \) is an integer. At these points, \( \csc x \) is undefined.
4. Therefore, \( \csc x \) is continuous on intervals between these points, specifically on intervals of the form \( (n\pi, (n+1)\pi) \) for any integer \( n \).
5. In conclusion, the function \( g(x) = \csc x \) is continuous on the intervals \( (n\pi, (n+1)\pi) \) for all integers \( n \), excluding the points where \( x = n\pi \).