What are non differentiable points for a graph?

1 Answer
Calculus V.
Jul 24, 2014

Since a function that is differentiable at a is also continuous at a, one type of points of non-differentiability is discontinuities .

On the other hand, if the function is continuous but not differentiable at a, that means that we cannot define the slope of the tangent line at this point. This can happen in essentially two ways:
1) the tangent line is vertical (and that does not have a slope)
2) the difference quotient (f(x)-f(a))/(x-a) whose limit at a defines the derivative has two different one-sided limits at a, resulting in two half-tangents. We call this situation a "cusp".
See this video on differentiability for details and pictures.

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