Continuity of a Piecewise Function2

Prove that the function $f(x)$ is continuous or discontinuous at $x = 2$ and $x = 3$ using the given piecewise function: $f(x) = \sqrt{x+2}$ when $x < 2$, $f(x) = x^2 - 2$ when $2 \leq x < 3$, $f(x) = 2x + 5$ when $x \geq 3$.

This problem explores the continuity of a piecewise function at specific points. To determine whether a function is continuous at a given point, you need to evaluate the left-hand limit, right-hand limit, and the function's value at that point. If all three values are equal, the function is continuous at that point; otherwise, it is discontinuous.

For the given piecewise function, consider how the different pieces of the function behave as x approaches the points of interest, 2 and 3, from either side. The problem requires you to assess continuity at these points by equating the relevant limit values. Each piece of the function captures various elementary function types such as square root and polynomial functions. Understanding the properties of these functions, especially around these critical points, is essential.

Continuity in piecewise functions is a critical concept in real analysis as it often involves combining different functional behaviors at specified junctions. Tackling this problem reinforces your ability to analyze such transitions and apply rigorous definitions of continuity in a structured manner, setting the foundation for more complex topics like differentiability and integration in functions.