For a cdf, $F(F^{-1}(u)) \geq u$ and $F^{-1}(F(x)) \leq x$. When does strict inequality apply?

Solution 1:

That looks correct.

To summarize, if the CDF is invertible, then your identities hold with equality. If it is not invertible, it is either because it is not injective or not surjective.

If it is not injective (e.g. it is constant over some interval), it doesn't have a left inverse so that $F^{-1}\circ F(x)\leq x$ should generally not hold with equality for some $x\in \mathbb{R}.$

If it is not surjective (e.g. it has jump discontinuities or atoms), it doesn't have a right inverse so that $F\circ F^{-1}(u)\geq u$ should generally not hold with equality for some $u\in [0,1].$