kolley said:Homework Statement
k(x)=x2*[1/x] for 0<x≤1
k(x)=0 for x=0
Find where k(x) is differentiable and find the derivative
Homework Equations
The Attempt at a Solution
I know that it is differentiable for all ℝ\Z on (0,1], but I am unsure how to find the derivative for this problem.
kolley said:Yes sorry that was a typo, should be (0,1). So would I set k=[1/x], which would make f(x)=x2*k
which would imply that f'(x)=2xk
Is this what you mean?
kolley said:If it's equal to an integer then it would not be differentiable.
kolley said:Because it's discontinuous at all integers.
The Differentiable Greatest Integer Function, also known as the floor function, is a mathematical function that rounds down a given real number to the nearest integer. It is denoted by ⌊x⌋ or [x].
No, the Differentiable Greatest Integer Function is not continuous. It has a jump discontinuity at every integer value, as the function abruptly changes from one integer to the next.
Since the Differentiable Greatest Integer Function is not continuous, it does not have a derivative in the traditional sense. However, it can be approximated by a piecewise function using the difference quotient.
The Differentiable Greatest Integer Function is often used in computer programming to truncate decimal numbers and to simulate rounding down in calculations. It is also used in economics and finance to model discrete changes in quantities.
The Differentiable Greatest Integer Function is a non-increasing function, meaning that as the input increases, the output decreases or remains the same. It is also a step function, with a constant value between integers. Additionally, it is a discontinuous function with infinitely many jump discontinuities.