What are non differentiable points for a graph?

Answer 1

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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Answer 2

Non-differentiable points on a graph occur where the function does not have a well-defined derivative. These points typically include corners, cusps, jumps, and vertical tangents. Additionally, non-differentiable points can occur at discontinuities, such as points where the function is not continuous or where there are vertical asymptotes. These points represent places where the slope of the function is undefined or discontinuous, making it impossible to calculate a derivative at those points.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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