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When we say a function is smooth? Is there any difference between smooth function and continuous function? If they are the same, why sometimes we say f is smooth and sometimes f is continuous? Please help me. Thanks.

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    $\begingroup$ "smooth" means (at least) "continously differentiable". Sometimes more (even infinite number of) derivatives are required to be continuous. $\endgroup$ – njguliyev Aug 20 '13 at 16:59
  • $\begingroup$ @njguliyev, not to nitpick but I think it's relatively common to call Lipschitz continuous ODEs "smooth" - being just smooth enough for existence and uniqueness of solutions. $\endgroup$ – alancalvitti Aug 20 '13 at 17:08
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A function being smooth is actually a stronger case than a function being continuous. For a function to be continuous, the epsilon delta definition of continuity simply needs to hold, so there are no breaks or holes in the function (in the 2-d case). For a function to be smooth, it has to have continuous derivatives up to a certain order, say k. We say that function is $C^{k}$ smooth. An example of a continuous but not smooth function is the absolute value, which is continuous everywhere but not differentiable everywhere.

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A smooth function is differentiable. Usually infinitely many times.

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    $\begingroup$ ... or at least as often as we need it. $\endgroup$ – Hagen von Eitzen Aug 20 '13 at 17:14
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A smooth function can refer to a function that is infinitely differentiable. More generally, it refers to a function having continuous derivatives of up to a certain order specified in the text. This is a much stronger condition than a continuous function which may not even be once differentiable.

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Smooth implies continuous, but not the other way around. There are functions that are continuous everywhere, yet nowhere differentiable.

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A smooth function is a function that has continuous derivatives up to some desired order over some domain. A function can therefore be said to be smooth over a restricted interval such as or . The number of continuous derivatives necessary for a function to be considered smooth depends on the problem at hand, and may vary from two to infinity. A function for which all orders of derivatives are continuous is called a C-infty-function.

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Take $f(x) = x|x|$ it is smooth, and now consider $g(x) = x^3$, this other function is also smooth. However, $g(x)$ is much smoother than $f(x)$ because derivitive of $f(x)$. You can argue that $g(x)$ is infinitely many times smooth. All polynomials belong to $C^\infty$ meaning they are infinitely many times differentiable and are smooth.

However, $h(x) = |x|$ is not smooth, because it has corner. Please note that all three functions, $f(x)$, $g(x)$, and $h(x)$ are continous.

Here is how $f(x)$ looks like:

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Here is how the derivative of $f(x)$ looks like:

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Here is the second derivate of $f(x)$, as you can see its second derivative is not even continuous:

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Here is the graph of $g(x)$:

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Here is graph of $\frac{d f(x)}{dx} = 3x^2$:

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Here is graph of $\frac{d^2 g(x)}{dx^2} = 6x$:

enter image description here

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  • $\begingroup$ If a function $f$ is smooth, then can I suppose that $f$ is increasing ou decreasing? At least in some interval? $\endgroup$ – Guilherme de Loreno Oct 9 '20 at 0:52
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Consider a sequence in $\mathbb{R}$ say $\{x_n\}_{n \in \mathbb{N}}$, which is continuous in $\mathbb{R}$. Usually we do not say it a smooth function.

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