This is a riff of my answer to a similar question on Math SE. It seems to better fit to this question anyway.
I mainly argue that it is highly unlikely that you will find such an example.
To this purpose, let’s look at your steps and what they would imply in practical situations:
- Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics.
Wrong statements from pure mathematics are made for a reason, namely that they hold in most cases the mathematician in question (and their peers, etc.) can think of.
Since mathematicians are usually good of thinking about relevant cases, this means that the cases in which such a statement fails are few in number as compared to those where it holds.
This alone makes it unlikely that such a case matters in application.
And that’s not even speaking of the several cases where the counter-example does not matter in reality at all.
In cases like “every continuous function is ‘mostly’ differentiable” this may be less obvious.
I address this separately below.
- That wrong statement was correctly applied to some other field, or somehow used in calculations etc.
Good science doesn’t just apply mathematics in a fire-and-forget sense.
It involves a lot of practical tests, double-checking, simulations, checking known cases, thinking about mathematical conditions and limitations, etc.
This is usually not because the underlying mathematics may be wrong, but because it may be applied for the wrong reason, etc.
Either way, these safety checks should also find problem due to wrong underlying mathematics.
Again, this may fail to find the error, but it’s unlikely.
Moreover, in a considerable amount of cases, the application may happen the other way around:
A scientists makes an empirical observation and then searches for the mathematics that allows to prove it.
Now, to allow for a (true) empirical observation, the mathematical statement must hold in a considerable number of relevant cases, which makes it unlikely to fail.
Of course, a lot of things may go wrong in this case, but then it’s a case of wrongly applied mathematics, lack of scientific rigour, etc.
- The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering.
This is mostly like step 2: Engineering again involves a lot of testing, etc. that should detect the problem.
Conclusion
So, for lack of rigour actually causing some real-life damage, a lot of unlikely things must coincide.
You can somewhat reduce the number of required coincidences by looking at one of those few cases, where mathematics is applied very directly in a case which does not allow for extensive testing (say space travel) though.
Appendix: What about fractals?
Now you may say: “Fractals are ubiquitous in nature and a counter-example to every continuous curve being ‘mostly’ differentiable”. (This other answer is based on this.) From this you may conclude that my argument that counter-examples to wrong statements are sparse in reality does not hold.
However, applying the entire notion of continuity or differentiability to non-atomic objects is already artificial in a way that should be clear to everybody applying mathematics.
Real fractures, coast lines, romanesco etc. are made of grains, cells, atoms, and similar, which in turn ultimately are not continuous in a way that translates to the macroscopic object.
Mind that this is not about the question whether reality is continuous or differentiable, but about how this manifests or can be applied.
For example, if you work with a derivative, you do not care about it for its own sake, but a physical meaning connected to it.
This physical meaning does not persist all the way down.
If you work on the assumption that your object has a physically meaningful derivative at these scales, you are making a scientific mistake anyway.
From another perspective, there are no fractals in the mathematical sense in reality.
(Fractals are “only” a very good way of modelling certain natural phenomena, the same way that continuity, derivatives, etc. are.)
For every realistic approximation of a non-differentiable curve in a macroscopic object, you will find an equivalent approximation of a differentiable curve.
So, to summarise, if somebody is really building something based on the assumption that continuous curves are “mostly” differentiable, they make at least one other, scientific mistake anyway.
