Is the Weierstrass function differentiable?
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
Does every continuous function have a derivative?
The Fundamental Theorem of Calculus tells us that every continuous function is the derivative of something, but there are many functions which are not continuous, and not derivatives. There are also some functions which are not continuous, but they are still derivatives.
Can a non continuous function be differentiable?
We see that if a function is differentiable at a point, then it must be continuous at that point. There are connections between continuity and differentiability. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .
Is Weierstrass function periodic?
It is periodic with period 2π. You can see it’s pretty bumpy. So bumpy, in fact, that it’s not differentiable anywhere.
Can you integrate the Weierstrass function?
The antiderivative of the Weierstrass function is fairly smooth, i.e. not too many sharp changes in slope. This just means that the Weierstrass function doesn’t rapidly change values (except in a few places). integrals, unlike derivatives, are highly insensitive to small changes in the function.
What are the examples of non differentiable functions?
Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.
How do you know if a function is non differentiable?
A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case.
Is derivative continuous if function is continuous?
A differentiable function is necessarily continuous (at every point where it is differentiable). It is continuously differentiable if its derivative is also a continuous function.
Can the derivative of a non continuous function be continuous?
Can discontinuous functions have derivatives?
The differentiability theorem states that continuous partial derivatives are sufficient for a function to be differentiable. It is possible for a differentiable function to have discontinuous partial derivatives. An example of such a strange function is f(x,y)={(x2+y2)sin(1√x2+y2) if (x,y)≠(0,0)0 if (x,y)=(0,0).