limit continuity concept question?

Question

i still seem to be struggling with limits here is the latest problem i don't understand

the question is:

The gravitational force exerted by earth on a unit mass at distance r from the center of the planet is

f(r) = GMr/R^3 if r GM/r^2 if r>or= to R
...where M is the mass of earth, R is its radius, and G is the gravitational constant. Is f a continuous function of r?


this is a piece wise fxn if that isn't clear.

Answer 1

As written, it -is- continuous since r/R^3 = 1/r² when r = R.
But at r=0 there is a discontinuity.

Doug

Answer 2

This is continuous. When r = 0 the function is GHr/R^3, which is continuous at 0 and for all r < R. When r is in the denominator r >=R, so r NE 0, and the function is still continuous when r > R. Finally, when r = R, both GMr/R^3 and GM/r^2 have the same value; this shows that the left- and right-hand limits are the same at r = R, so the function is continuous at r = R.