In the question Smooth transition between two lines (2d) there is an example of a composite curve which has a point where it is non-smooth.
In general, what is the name for that transition point?
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The most general concept/word would be singularity.
In this case, I might call it a "corner", a concept which can be made formal with the concept of a "smooth manifold with corners" as mentioned in this MSE answer, for instance.
Depending on the context/exact situation, something like this may be called a cusp or crunode.
