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For example $f(x) = 1_{\{x < 0\}}x + 1_{\{x > 0\}}x^2$. Before and after $x = 0$ there are two different functions. Is there any name for such points?

I am aware of critical points, extremum points, stationary points, inflection points

Jens Renders
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Kate
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    I doubt it's relevant in this context, but perhaps "bifurcation point"? In ODEs that's used to indicate when the solutions to ODEs/stability of ODEs to change their behaviour – peek-a-boo May 24 '20 at 10:30
  • @peek-a-boo Actually I think its good, I can accept this as an answer – Kate May 24 '20 at 10:38
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    It is a point of non-differentiability, but I feel this is not what you are getting at. I think you are talking about the fact that you have a piecewise defined function which is given by a different formula at both ends of that point. But this is hard to pin down because what is and what is not allowed as a formula? – Jens Renders May 24 '20 at 10:55
  • It is not a bifurcarion point, as @peek-a-boo also points out himself, so that should not be put as an anwer. It is just an interesting comment. – Jens Renders May 24 '20 at 10:58
  • @JensRenders: piecewise must be qualified. This gives an opportunity to specify what is allowed. E.g. piecewise cologarithmic. :) –  May 24 '20 at 11:39

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Your function is $C^0$ (continuous) but not $C^1$ (the derivative at $x=0^-$ is different from the derivative at $x=0^+$: as @Jens Renders pointed out it is just a point of non-differentiability (see the first example here).

In the language of phase transitions (according to the classification by Ehrenfest) you may say that at $x=1$ there is a first order phase transition (if you interpret your $f$ as a thermodynamic potential, like the free energy).

Quillo
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If you mean to express that the function has a different analytical expression in different parts of its domain, you should call it piecewise.

In the given example, it would be piecewise quadratic (accepting that the linear part is a "degenerate" quadratic).

Also note that your function has an angular point at the origin.