Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.
$g\left(x\right)=\cos \left(e^x\right)$

Verified step by step guidance

1. Step 1: Identify the components of the function g(x) = cos(e^x). The function is composed of an exponential function e^x and a trigonometric function cos(x).
2. Step 2: Determine the continuity of the inner function e^x. The exponential function e^x is continuous for all real numbers x.
3. Step 3: Determine the continuity of the outer function cos(x). The cosine function is continuous for all real numbers.
4. Step 4: Use the composition of continuous functions theorem. Since both e^x and cos(x) are continuous for all real numbers, their composition g(x) = cos(e^x) is also continuous for all real numbers.
5. Step 5: Conclude that g(x) = cos(e^x) is continuous on the interval (-∞, ∞). There are no finite endpoints to consider for right- or left-continuity.

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 a function does not have breaks, jumps, or asymptotes.

Endpoints and Continuity

When analyzing the continuity of functions on closed intervals, special attention must be given to the endpoints. A function can be left-continuous at the left endpoint and right-continuous at the right endpoint. Understanding how a function behaves at these endpoints is essential for accurately describing its continuity over the entire interval.

Behavior of Composite Functions

The function g(x) = cos(e^x) is a composite function, where the continuity of g depends on the continuity of both the cosine function and the exponential function. Since both functions are continuous everywhere, g(x) is also continuous for all real numbers. Recognizing how the continuity of inner and outer functions affects the overall function is vital in calculus.