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Is that really fundamental? Maybe for studies in pathological real functions.

But in realistic functions relevant to our actually universe, these pathological cases aren't important.

I would argue that understanding the pathological behavior in something is critical to developing an accurate intuition for it, yes. These cases don't show up often, but when it comes to having a good sense of smell for when part of a proof is flawed, it really helps to have that olfactory memory.

Aside from that, understanding counterexamples teaches you to understand the definitions and theorems better. Which matters for proving future results.

Fractals and Dirac delta functions both have somewhat pathological properties at times, and both pop up as approximations of real systems in physics.

I would personally not consider it fundamental either though, more of a “let’s cross that bridge when we get to it” problem.

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