Table of Contents
How can you tell if a function is differentiable?
A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean? It means that a function is differentiable everywhere its derivative is defined. So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable.
Does complex differentiable imply continuous?
If a function is differentiable then it’s also continuous. This property is very useful when working with functions, because if we know that a function is differentiable, we immediately know that it’s also continuous.
Can function be differentiable but not continuous?
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 .
Where is a complex function differentiable?
The function f is complex-differentiable at an interior point z of A if the derivative of f at z, defined as the limit of the difference quotient f′(z)=limh→0f(z+h)−f(z)h f ′ ( z ) = lim h → 0 f ( z + h ) − f ( z ) h exists in C.
What is the derivative of a complex number?
The notion of the complex derivative is the basis of complex function theory. The definition of complex derivative is similar to the derivative of a real function. However, despite a superficial similarity, complex differentiation is a deeply different theory. f′(z0)=limz→z0f(z)−f(z0)z−z0.
What does it mean for functions to be differentiable?
In other words, it’s the set of all real numbers that are not equal to zero. So, a function is differentiable if its derivative exists for every x -value in its domain . Let’s have another look at our first example: f ( x) = x 3 + 3 x 2 + 2 x. f ( x) is a polynomial, so its function definition makes sense for all real numbers.
Why is x = 0 not differentiable at a point?
To be differentiable at a certain point, the function must first of all be defined there! As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is “heading towards”. So it is not differentiable.
Is the derivative of F differentiable everywhere?
So, the derivative of f is f ′ ( x) = 3 x 2 + 6 x + 2. This derivative exists for every possible value of x! So, f is differentiable: we can find it’s derivative everywhere! Most of the above definition is perfectly acceptable.
What does a differentiable function look like under a microscope?
If you were to put a differentiable function under a microscope, and zoom in on a point, the image would look like a straight line. When you zoom in on the pointy part of the function on the left, it keeps looking pointy – never like a straight line.