- #1
jobsism
- 117
- 0
Are |x| and [x] differentiable anywhere? If so, what're their derivatives?
mathman said:This can be written as x/|x| for x ≠ 0.
|x| differentiability refers to the differentiability of the absolute value function, where the derivative exists at all points except for 0. [x] differentiability refers to the differentiability of the greatest integer function, where the derivative exists at all points except for integers.
Differentiability is a property of a function where the derivative exists at every point in the domain. This means that the function is smooth and has a well-defined slope at every point.
A function is differentiable at a specific point if the left-hand and right-hand derivatives at that point are equal. This means that the function is continuous and has a well-defined slope at that point.
If a function is differentiable, it means that it is continuous and has a well-defined slope at every point in its domain. This allows us to use calculus to analyze the function and make predictions about its behavior.
Yes, both |x| and [x] are continuous functions. This means that they have no breaks or gaps in their graphs and they can be drawn without lifting the pen from the paper.