Prove $F$ is a continuous function of $r$ or not

calculuspiecewise-continuity

I have this one problem in my Calculus 1 assignment that I'd like to ask.
The gravitational force exerted by the planet Earth on a unit mass at a distance $r$ from the center of the planet is
$$F(r) = \begin{cases} \frac{GMr}{R^3} & r < R \\ \frac{GM}{r^2} & r \ge R \end{cases}$$

where $M$ is the mass of Earth, $R$ is its radius, and $G$ is the gravitational constant. Is $F$ a
continuous function of $r$?
I have no idea where to start and I'm completely stuck here.

Best Answer

Hint: Since both pieces of the piecewise function are continuous functions in their respective domains of definition, it is just required that the continuity be checked at the point $r=R$, which would be done by checking for the following conditions:

  • $\lim_{r\to R} F(r)$ exists
  • $F(r=R)$ exists
  • $F(r=R)=\lim_{r\to R} = F(r)$
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