Sage-Tips

Under what conditions is a function not 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.

What kinds of functions are not differentiable?

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.

Is the complex conjugate differentiable?

Conjugation is a reflection so it flips orientation, therefore it cannot be differentiable at any point in the complex sense.

What is the first derivative of a function?

The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant. Because of this definition, the first derivative of a function tells us much about the function. If is positive, then must be increasing. If is negative, then must be decreasing.

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.

Is z 2 differentiable everywhere?

Example: The function f (z) = |z|2 is differentiable only at z = 0 however it is not analytic at any point.

What is the difference between real and complex differentiability?

Remark – Real/Complex-Differentiability. We call this operator real-differential to avoid any ambiguity with the complex-differential. The definitions of both operators are identical, except that the complex-differential is required to be a complex-linear operator when the real-differential is only required to be real-linear.

Are all real-valued functions of a real variable differentiable?

This similar result doesn’t hold in the context of real analysis: there are some real-valued functions of a real variable that are differentiable and whose derivative is not continous 1. We mention this property now because we will use it to simplify the statements of some results of the current and subsequent chapters.

Is the derivative of a holomorphic function continuous?

‌ The derivative of a holomorphic function is always continuous. This similar result doesn’t hold in the context of real analysis: there are some real-valued functions of a real variable that are differentiable and whose derivative is not continous 1.

What is the difference between real and complex differential operators?

The definitions of both operators are identical, except that the complex-differential is required to be a complex-linear operator when the real-differential is only required to be real-linear. Proof – Uniqueness. Proof – Uniqueness.