Question: Continuous at x = c if…

Answer: f(c) exists, limx→cf(x) exists, and it = f(c) coming from left OR right side

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Question: Removable discontinuity

Answer: the limit exists but f(c) does not. this is a PATCHABLE discontinuity; just change the equation at point x = c.

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Question: Piece-wise continuous

Answer: eg rational functions

made of several disconnected continuous functions

figure out if it’s continuous by see in if the critical values’ limits match up

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Question: What if f(x) exists but f’(x) doesn’t?

Answer: it’s NON-DIFFERENTIABLE (the point has no slope)

corner, cusp, vert tangent, or discontinuity

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Question: Corner

Answer: non-differentiable; where one-sided derivatives differ,

eg. f(x) =|x|

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Question: Cusp

Answer: non-differentiable: secant lines approach −∞ & ∞. Extreme corner. Eg. f(x) =x^(2/3)

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Question: Vertical tangent

Answer: non-differentiable: secants both approach either −∞ or ∞, e.g. f(x) = ³√x

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Question: Discontinuity

Answer: non-differentiable: on one or both sides, the limit does not exist.

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Question: Differentiability implies continuity . . . .

Answer: . . . . but continuity does NOT imply differentiability!

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Question: Local linearity

Answer: differentiability implies this. graph resembles own tangent if you zoom in enough (looks linear).

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Question: IVT

Answer: Intermediate Value Theorem: if f(x) is continuous on a and b, your graph hits EVERY y value between a and b.

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