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Smooth Function: Simple Definition, Examples, C^n

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Types of Functions >

A smooth function is just like the name sounds: it’s a function that travels without any drop offs, jumps or other strange behavior that would make it not differentiable. More specifically, the function is differentiable up to some desired point.


That desired point is called the “class“. It is denoted:
Cn
Where “n” is the order.

An order is just the number of derivatives. For example, a first derivative is order 1, a fourth derivative is order 2, and a function that can be differentiated an infinite number of times is order ∞.


Smooth functions include exponential functions, trigonometric functions and many others.

Smooth Function on an Open Interval

A real-valued, smooth function of class Cn, defined on an open interval has the following characteristics:

  • The class (n) is in the set of natural numbers; this is written as n ∈ ℕ (Natural numbers are counting numbers, i.e. whole, non-negative numbers).
  • The function is defined on the open interval (a, b). An “open interval” doesn’t contain the endpoints.
  • All derivatives exist up to order n, on the stated interval. For example, a smooth function of class C2 has both a first derivative and a second derivative. If all derivatives exist, the function is called infinitely smooth or infinitely differentiable.
  • The derivatives are continuous. In other words, its derivative is a continuous function.

Smooth Function on a Closed Interval [a, b]

Smooth functions can also be defined on a closed interval [a, b]. The definitions are the same as that for open intervals, except that the closed interval includes endpoints. The function ends abruptly at the endpoints of a closed interval, so the function can’t be described as “smooth” at that exact point without additional information about what happens at “a” and “b”.

Specifically, a smooth function of class Cn, defined on [a, b], has continuous, one-sided derivatives at point a and b.

  • For point a, the one-sided derivative from the right exists.
  • For point b, the one-sided derivative from the left exists.

These derivatives must be the same order (i.e. both must be differentiable the same x number of times).

smooth function

This smooth function has one sided derivatives from both the left, and right.


Note that if the function is half closed: (a, b] or [a, b), then only the closed side will have a one-sided derivative.

References

Shikin, E. (2014). Handbook and Atlas of Curves. CRC Press.

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Stephanie Glen. "Smooth Function: Simple Definition, Examples, C^n" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/smooth-function/
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