Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 40

Check if $F$ is a continuous function of $r$ for the gravitational force $
\displaystyle
F(r) = \left\{
\begin{array}{c}
\frac{GMr}{R^3} & \text{ if } & r < R\\
\frac{GM}{r^2} & \text{ if } & r \geq R
\end{array}
\right.
\quad
$
exerted by the earth on a unit mass at a distance $r$ from the center of the planet.


Where $M$ is the mass of the earth, $R$ is its radius and $G$ is the gravitational constant.

Based from the definition of continuity,
The function is continuous of at a number if and only if the left and right hand limits of the function at the same number is equal. So,



$
\begin{equation}
\begin{aligned}

\lim \limits_{r \to R^-} \frac{GMr}{R^3} &= \lim \limits_{r \to R^+} \frac{GM}{R^2}\\
\frac{GM\cancel{(R)}}{R^{\cancel{3}}} & = \frac{GM}{(R)^2}\\
\frac{GM}{R^2} &= \frac{GM}{R^2}

\end{aligned}
\end{equation}
$


Therefore,
$F$ is a continuous function of $r$.