The gravitational force exerted by the earth on an object...

The gravitational force exerted by the earth on an object having mass $m$ that is a distance $r$ from the center of the earth is $$g(r)=\left\{\begin{array}{ll} \frac{G M m r}{R^{3}}, & \text { if } r<R \\ \frac{G M m}{r^{2}}, & \text { if } r \geq R \end{array}\right.$$ Here $G$ is the gravitational constant, $M$ is the mass of the earth, and $R$ is the earth's radius. Is $g$ a continuous function of $r ?$

The gravitational force exerted by the earth on an object having mass $m$ that is a distance $r$ from the center of the earth is
$$g(r)=\left\{\begin{array}{ll}
\frac{G M m r}{R^{3}}, & \text { if } r<R \\
\frac{G M m}{r^{2}}, & \text { if } r \geq R
\end{array}\right.$$
Here $G$ is the gravitational constant, $M$ is the mass of the earth, and $R$ is the earth's radius. Is $g$ a continuous function of $r ?$

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Key Concepts

Continuity

Continuity is a fundamental concept in calculus which states that a function is continuous at a point if the function's value at that point equals the limit of the function as the input approaches that point from both directions. In other words, there should be no sudden jumps, holes, or breaks at that point, which is critical when evaluating functions that are defined in different regions of their domain.

Piecewise Functions

Piecewise functions are those that are defined by different expressions for different parts of their domain. Analyzing piecewise functions often requires checking the behavior at the boundaries where the definition changes to ensure that the function is well-behaved and, in problems like this, continuous, which involves verifying that the limits from either side at the boundary are equal.

Newton's Law of Gravitation

Newton's law of gravitation describes the gravitational force between two masses and states that the force is proportional to the product of the two masses and inversely proportional to the square of the distance between them. This law is applied in different forms depending on the distribution of mass, such as inside a spherically symmetric mass distribution versus outside it, and is key to understanding forces like the gravitational pull exerted by the Earth.

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