Reasons a f'(a) COULD FAIL TO EXIST

Three reasons why f'(x) could fail to exist at the point x=a

1. The function is discontinuous at a point "a"

2. The function has a vertical tangent at "a".

3. The function has a "corner point" at "a"

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The function is discontinuous at a point.​

The function could have a "removable discontinuity", a "vertical asymptote", a "jump discontinuity" or simply be "undefined" at a x="a". If f(a) DNE (does not exist), then f'(a) does not exist either. If f(a) is discontinuous, then f'(a) fails to exist.

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f(a) has a​ vertical tangent at "a". For the graph at the left, the vertical tangent is at x=0​

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The function has a "corner" point, like the one shown at the right at x=3​

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