Smooth function
High school
Consider a function, its derivative, the derivative of that, and so on. The function is smooth if you can take as many derivatives of it as you like, and each derivative is continuous.
Undergraduate
Consider a function on an open set, and its derivatives of each order. The function is smooth if it has continuous derivatives of every order.
Examples
Examples
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$x \mapsto \sin x$
High school
$x \mapsto \sin x$ is smooth, because its derivatives $\cos x$, $-\sin x$, $-\cos x$, $\sin x$, … repeat and all exist.
Non-examples
Things that may be mistaken for examples.
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$f(x) = \abs{x}$
High school
$f(x) = \abs{x}$ is not smooth, because it is not even differentiable at $0$.
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$x \mapsto x \abs{x}$
High school
$x \mapsto x \abs{x}$ is not smooth, because its derivative $x \mapsto 2 \abs{x}$ is not differentiable at $0$.
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$f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$
Undergraduate
$f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$, is not smooth, because $f^{\prime}(x) = 2x \sin(1/x) - \cos(1/x)$ has no limit as $x \to 0$.
Notes
Some authors use “smooth” loosely to mean “continuously differentiable” or “as differentiable as needed.” Check the author’s definition.