"If a periodic function is continuous and nonconstant, then it has a least period, and all other periods are positive integer multiples of the least period."
I think my naive intuition is modelling "periodic function" as "continuous and periodic". Basically, I'm not exercising the the full freedom of the "it's periodic" concept.
There's a good Feynman anecdote on this (from Surely You're Joking...):
------
"I had a scheme, which I still use today when somebody is explaining something that I'm trying to understand: I keep making up examples.
For instance, the mathematicians would come in with a terrific theorem, and they're all excited. As they're telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball)-- disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on.
Finally they state the theorem, which is some dumb thing about the ball which isn't true for my hairy green ball thing, so I say "False!" [and] point out my counterexample."
------
So I think it's useful to consider the most extreme thing which meets your criteria, instead of a "representative" example.
To come back to tech I think a similar mindset is also useful for things like system failure mode analysis. "Yes, but what if that switch dies at the same time...?"