# smooth functions or continuous

When wesay 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?

• "smooth" means (at least) "continously differentiable". Sometimes more (even infinite number of) derivatives are required to be continuous. Aug 20, 2013 at 16:59
• @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. Aug 20, 2013 at 17:08

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.

A smooth function is differentiable. Usually infinitely many times.

• ... or at least as often as we need it. Aug 20, 2013 at 17:14

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

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.

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.

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:

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

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

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

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

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

• If a function $f$ is smooth, then can I suppose that $f$ is increasing ou decreasing? At least in some interval? Oct 9, 2020 at 0:52

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.