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Definition: A function f is differentiable at a if f'(a) exists. it is differentiable on an open interval (a, b) [or (a, ∞) or (-∞, a) or (-∞, ∞)]if it is differentiable at every number in the interval.

Example: Where is the function f(x) = |x| differentiable?

Answer:

1. f is differentiable for any x > 0 and x < 0.

2. f is not differentiable at x = 0.

That's mean that the curve y = |x| has not a tangent at (0, 0).

Thus, both continiuty and differentiability are desirable properties for a function to have. These properties are related.

Theorem: If f is differentiable at a, then f is continuous at a.

The converse theorem is false, that is, there are functions that are continuous but not differentiable. (As we saw at the example above. f(x) = |x| is contionuous at 0, but is not differentiable at 0).

The three ways for f not to be differentiable at aare:

a) if the graph of a function f has a "corner" or a "kink" in it,

b) a discontinuity,

c) a vertical tangent

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