In most maths textbooks, proofs by induction prove a statement $P_n$ where $n$ usually is in the natural numbers (although I understand that it can be in any discrete collection as long as you prove your base case(s) and prove all cases imply the next is true).

However, I was wondering if there was some form of induction which could prove statements in the Reals, or even in the Complex numbers:

First prove the base case, $P_c$.

Then show that if $P_k$ is true, then $P_{k+a}$ is also true $\forall \,a \in \mathbb{R}$.

OR

Show that if $P_k$ is true, then $P_{k+a}$ is also true $\forall \,a,l \in \mathbb{R},\, a < l$ where $l$ is some kind of 'upper limit' — 1 for example. (For example, if $P_{k}$ is true then we could show that $P_n$ is also true for all of the values which are between 0 and 1 above $k$ on the number line).

(Note, that this could be generalised to strong induction, and in the complex areas you'd be proving that a case implies a certain region of cases near that case in the Complex plane)

If this kind of induction does exist, then what kind of applications does it have in maths? What kind of proofs need this induction?