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I am having a bit of trouble understanding why a derivative does not exist at points where the function has a sharp jump. Is it because the slope of the tangent line as we approach the point from the left and right direction changes sign? My book does not go into detail and I looked online and ended up confusing myself even more as some sites say that you could draw infinite number of tangents through the point or some other explanation and I am unable to convince myself of the exact reason why a sharp point is a problem.

Looking forward to your help. Many thanks,

Luca