- #1
sphyics
- 102
- 0
f(x)= mod x is this function differentiable at all points other than 0.
yes its absolute value of x :)CompuChip said:Did you mean "f(x) = |x|", the absolute value of x?
In that case, the answer is yes: you can easily see this by showing that it is equivalent to x for x > 0 and -x for x < 0. To prove that it is non-differentiable at x = 0, you need to do a little work, but that's still straightforward.
gb7nash said:I think the OP meant modulus, which is another term for absolute value.
No, mod x (or absolute value of x) is not differentiable at all points. It is not differentiable at the point where x=0, because the derivative is undefined at this point.
Yes, we can use the limit definition to determine if mod x is differentiable at a point. If the right-hand and left-hand limits of the derivative exist and are equal at a point, then the function is differentiable at that point.
The derivative of mod x is not a single value, as it is not differentiable at the point x=0. However, the derivative of mod x at any other point is equal to either 1 or -1, depending on the sign of x.
Mod x is not differentiable at x=0 because the function is not smooth at this point. The left-hand and right-hand limits of the derivative do not exist and are not equal, making the derivative undefined at this point.
Yes, there are other functions that are not differentiable at all points. Some examples include the absolute value of x raised to an odd power, the step function, and the floor function. These functions have points where the derivative is undefined or discontinuous, making them not differentiable at those points.