Continuity of a Piecewise Function

Determine the value of the constant $C$ that will make the piecewise function $f(x) = \begin{cases} Cx + 3, & \text{if } x < 2 \\ 3x + C, & \text{if } x \geq 2 \end{cases}$ continuous at $x = 2$.

In this problem, you are given a piecewise function and asked to determine a constant that will ensure the function is continuous at a given point. Continuity of a function at a point means that the limit of the function as you approach the point from both sides is equal to the function's value at that point. For piecewise functions, checking continuity often involves ensuring that the different function definitions on either side of the point in question meet smoothly. Crucially, this means that you need to understand how limits work and ensure the left hand and right hand limits are the same and equal to the function's value at the point.

Students should be familiar with the epsilon-delta definition of a limit and continuity, even though this problem does not require its direct application. The problem is a practical exercise in applying the definition of continuity and involves basic algebra to solve for the unknown constant. It highlights the importance of having a clear conceptual understanding of what continuity means in a piecewise context where different expressions can govern function behavior on different intervals.