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In general, what non-differentiable functions have subgradient? Which non-differentiable functions do not have subgradient?

Does a function have to be convex for subgradient to be defined for that? How about $$f(x) = \sqrt{|x|}?$$

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    As far as I know, subgradient is only defined for convex functions. And the definition is clear enough. I am not getting what yo mean by _does a function have to be convex for subgradient to be defined for that_ $$\;$$ It's a **definition** – 2017-01-23
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    I simply meant if subgradient is only defined for convex functions or non-convex too. – 2017-01-23
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    A function whose domain is a convex set _and_ has a subgradient at all points in its domain is convex. It is possible for 1) a function with a non-convex domain to have a subgradient on that domain; and 2) a function to have a subgradient on only a subset of its domain. $f(x)=\sqrt{|x|}$ has a subgradient only at the origin; specifically, $\partial f(0)=\{0\}$. – 2017-01-23
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    Practically, however, the notion of subgradients is of effectively no use for non-convex functions. – 2017-01-23
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    @MichaelGrant I see. Thanks! – 2017-01-23

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