Answer:
Yes the function is continuous.
Step-by-step explanation:
Given
[tex]F(r)=\frac{GMr}{R^{3}}(r< R)[/tex] and [tex]F(r)=\frac{GM}{r^{2}}(r\geq R)[/tex]
For a function to be continuous we should have
[tex]\lim_{r\rightarrow R^{-}}F(r)=\lim_{r\rightarrow R^{+}}F(r)[/tex]
Thus
[tex]\lim_{r\rightarrow R^{-}}F(r)=\frac{GMR}{R^{3}}=\frac{GM}{R^{2}}[/tex]
Similarly we have
[tex]\lim_{r\rightarrow R^{+}}F(r)=\frac{GM}{R^{2}}[/tex]
thus we can see that
[tex]\lim_{r\rightarrow R^{-}}F(r)=\lim_{r\rightarrow R^{+}}F(r)[/tex]
Hence the function is continuous.
By evaluating both pieces of the function in r = R, we will see that the function is continue.
The force F will only be continuous if in the limit r = R (where the piecewise function changes, let's say) we have the same value in both pieces of the function.
In this case, we have:
F(r) = (G*M r/R^3) if r < R
F(r) = G*M/r^2 if r > R
To evaluate the continuity, we need to evaluate both of these in r = R and see if we get the same thing.
For the first one we have:
F(R) = G*M*R/R^3 = G*M/R^2
For the second one we have:
F(R) = G*M/R^2
So yes, we got the same thing in both cases, meaning that F is a continuous function of r.
if you want to learn more about continuity, you can read
https://brainly.com/question/24637240
