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Statement: A function $f$ is continuous at a point $x=a$ if it has different finite (and real) left and right hand derivatives at that point.

I've tried finding counter examples but have not found any so far. This statement holds true for $f(x) = |x|$ at point $x=0$ as it has a finite right hand derivative $x=-1$ and a finite left hand derivative $x=1$, and it continuous at that point. I've come across the Weierstrass function but that function doesn't have right and left hand derivatives (at any point).

If this statement is true, how could I go about proving it?

Edit: I'm wondering if having a (different) finite left and right derivative at a point is a sufficient condition for a function to be continuous at that point. I'm not wondering if having a left or right derivative is a necessary condition for a function to be continuous (it isn't).

Tomyy
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    Concerning your question in the title the answer is yes. The existence of the RHS derivative at $a$ implies RHS continuity at $a$ while the existence of LHS derivative implies LHS continuity. Thus the function is continuous at $a.$ – Ryszard Szwarc Sep 06 '23 at 16:49

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