Can a function be differentiable but not continuous at a point?

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 .

Which of the functions are differentiable but not analytic?

Differentiability =⇒ Analyticity. Example: The function f (z) = |z|2 is differentiable only at z = 0 however it is not analytic at any point. Let f (z) = u(x, y) + iv(x, y) be defined on an open set D ⊆ C.

How do you check if a complex function is differentiable at a point?

‌ Let f:A⊂C→C. 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.

When a complex function is analytic?

A function is complex analytic if and only if it is holomorphic i.e. it is complex differentiable. For this reason the terms “holomorphic” and “analytic” are often used interchangeably for such functions.

Is the function differentiable at a point?

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.

What is analytic function in complex variable?

A function f(z) is said to be analytic in a region R of the complex plane if f(z) has a derivative at each point of R and if f(z) is single valued. If f(z) is analytic at a point z, then the derivative f (z) is continuous at z.

Is Z 2 complex differentiable?

(2) If f : C → C is differentiable everywhere and f(z) is real for all z ∈ C then f is a constant function. This follows from CR equation as v(x, y) = 0 for all x + iy ∈ C and hence all partial derivatives of v is also zero and hence the same is true for u. Thus the function f(z) = |z|2 is not differentiable for z = 0.

When a function is analytic at a point?

A function f(z) is said to be analytic in a region R of the complex plane if f(z) has a derivative at each point of R and if f(z) is single valued. A function f(z) is said to be analytic at a point z if z is an interior point of some region where f(z) is analytic.