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Continuity denotes that a function does not have any abrupt jumps, breaks, or holes. If you can draw a function on a graph without lifting your pen, it is continuous. For a function to be continuous at a point, three conditions must hold:
1. The function must be defined at that point.
2. The limit of the function as it approaches the point from both sides must exist.
3. The limit of the function as it approaches the point must equal the function's value at that point.

In this problem, we are given a piecewise function and asked to find the conditions under which it is continuous over its domain. Specifically, we need to ensure continuity at the point where the function's definition changes, which is at x = 4000.

A piecewise function is one that is defined by different expressions over different intervals. In this problem, the weight function, \(w(x)\), changes its form depending on whether \(x < 4000\) or \(x > 4000\). For \(x \leq 4000\), we have \(w(x) = Ax\), a linear function and for \(x > 4000\), \(w(x) = \frac{B}{x^2}\), a hyperbolic function.

Piecewise functions are often used to model situations where a certain rule does not apply across the entire domain but changes at specific points. This makes them suitable for real-world problems like this, where the weight changes based on the distance from the Earth’s center.

The concept of a limit is fundamental in calculus and crucial for understanding continuity. When we say the limit of \(f(x)\) as \(x\) approaches a value \(c\), we mean the value that \(f(x)\) gets closer to as \(x\) gets closer to \(c\).

• To find the limit from the left (\text{lim}_{x \to c^-}), we examine values of \(x\) that are less than \(c\).


• To find the limit from the right (\text{lim}_{x \to c^+}), we examine values of \(x\) that are greater than \(c\).

In this exercise, we need to ensure that the limit of \(w(x)\) as \(x\) approaches 4000 from the left (\(w(x) = Ax\)) equals the limit as \(x\) approaches 4000 from the right (\(w(x) = \frac{B}{x^2}\)). By setting these limits equal to each other, we ensure the function is continuous at \(x = 4000\), leading to our continuity condition: \ A \cdot 4000 = \frac{B}{4000^2}. Solving this equation helps us find the necessary relationship between the constants \(A\) and \(B\).