Consider the Initial Value Problem, $\frac{dP}{dt} = ((1 + f)^\frac{3}{5} (1 - f)^\frac{2}{5} - 1)P,\ P(0) = 100$, where $f$ is a real parameter between $0$ and $1$, inclusive. Its solution is $P = 100e^{(1\ +\ f)^\frac{3}{5} (1\ -\ f)^\frac{2}{5}t\ -\ t}$.
The discrete analog to this Initial Value Problem is $P[n + 1] = ((1 + f)^\frac{3}{5} (1 - f)^\frac{2}{5} - 1)P[n],\ P[0] = 100$. Its solution is $P = 100((1 + f)^\frac{3}{5} (1 - f)^\frac{2}{5} - 1)^n$. Note that $f$, and thus $P$, are still continuous; only the independent variable has been discretized.
I would expect the graphs of the two solutions to resemble each other; the former is a surface, and the latter should look like a parallel succession of slices of the same surface. However, this is the actual geometry...
There is some unrelated issue with improper scaling of $f$ in the second graph. However, it is still pretty clear that the latter is not a discrete version of the former. Why is it not?