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Concept# Semi-differentiability

Summary

In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.
In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.
Let f denote a real-valued function defined on a subset I of the real numbers.
If a ∈ I is a limit point of I ∩ and the one-sided limit
exists as a real number, then f is called right differentiable at a and the limit ∂+f(a) is called the right derivative of f at a.
If a ∈ I is a limit point of I ∩ and the one-sided limit
exists as a real number, then f is called left differentiable at a and the limit ∂–f(a) is called the left derivative of f at a.
If a ∈ I is a limit point of I ∩ and I ∩ and if f is left and right differentiable at a, then f is called semi-differentiable at a.
If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define a symmetric derivative, which equals the arithmetic mean of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not.
A function is differentiable at an interior point a of its domain if and only if it is semi-differentiable at a and the left derivative is equal to the right derivative.
An example of a semi-differentiable function, which is not differentiable, is the absolute value function , at a = 0. We find easily
If a function is semi-differentiable at a point a, it implies that it is continuous at a.

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Difference quotient

In single-variable calculus, the difference quotient is usually the name for the expression which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x + h) - x = h in this case). The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h).

Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative exists.

Gradient

In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) whose value at a point is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative.

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