Differentiability
A function is differentiable at a if its derivative exists there. A sharp corner, a vertical tangent or a break means it is not.
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Analogy
A smooth hill is easy to slope-check, but at a pointed roof peak there is no single slope.
Example
Right of 0 the slope is 1.Left of 0 the slope is -1.The slopes disagree.So f'(0) does not exist.
In practice
Many theorems need differentiability, and every differentiable function is continuous.
Practice
Flashcards
- Is |x| differentiable at 0?No. The left slope is -1 and the right slope is 1.
Questions people ask
Does continuity imply differentiability?
No. The absolute value function is continuous at 0 but has a sharp corner there, so it has no derivative. The reverse holds: if a function is differentiable at a point, it must be continuous at that point. So continuity is necessary for a derivative, but it is not enough.
How can I tell where a function is not differentiable?
Look for corners, cusps, jumps and vertical tangents. For |x|, the left and right slopes disagree at 0. For the cube root of x, the slope is infinite at 0. Algebraically, check that the one-sided derivatives exist and are equal. If they differ, or either is infinite, the derivative does not exist at that point.