What is the subgradient set of the convex function f(x) = |x| at the point x = 0?

### Why a generalised derivative is needed For a smooth convex function, the test for a minimum is that the derivative vanishes. That test is useless at a kink, where no derivative exists — and kinks are exactly where interesting minima often sit. The subgradient generalises the derivative to convex functions in a way that keeps the optimality test working. ### The definition A function `f` is convex when `f(t*x + (1-t)*y) <= t*f(x) + (1-t)*f(y)` for all points `x`, `y` and all `t` in `[0, 1]`. For such a function, a number `g` is a **subgradient of f at x0** when ``` f(x) >= f(x0) + g*(x - x0) for every x ``` The right-hand side is the line through the point `(x0, f(x0))` with slope `g`. The condition says that line lies weakly below the graph everywhere — it is a **supporting line**. The set of all subgradients at `x0` is the **subdifferential**, written `df(x0)`. It is always a closed convex set; for a convex function that is finite near `x0` it is non-empty. Where `f` is differentiable, the tangent line is the only line that supports the graph from below, so the subdifferential is the single number `f'(x0)`. The set-valued definition therefore extends the derivative rather than replacing it. ### Computing the subdifferential of |x| at 0 With `f(x) = |x|`, `x0 = 0` and `f(0) = 0`, the defining inequality becomes ``` |x| >= g*x for every real x ``` **Necessity.** Put `x = 1`: the condition gives `1 >= g`. Put `x = -1`: it gives `1 >= -g`, i.e. `g >= -1`. So any subgradient must satisfy `-1 <= g <= 1`. **Sufficiency.** Suppose `|g| <= 1`. Then for every `x`, `g*x <= |g*x| = |g|*|x| <= |x|`. The inequality holds, so every such `g` is a subgradient. Therefore the subdifferential is exactly the **closed** interval `[-1, 1]`. The endpoints belong to it: the line of slope `1` coincides with the right branch of the V and stays below the left branch, and symmetrically for slope `-1`. Geometrically, the subdifferential at the vertex of a V is the whole fan of lines you can wedge under the crease without poking through. At a point `x0 > 0` the only supporting line is the tangent of slope `1`, so `df(x0) = {1}`; for `x0 < 0` it is `{-1}`. The subdifferential is a single point wherever the function is smooth and expands into an interval exactly at the kink. ### The optimality condition The reason this construction earns its place is one clean theorem: for a convex `f`, a point `x*` is a **global** minimiser if and only if ``` 0 belongs to df(x*) ``` The proof is immediate from the definition: `0` being a subgradient means `f(x) >= f(x*) + 0*(x - x*) = f(x*)` for every `x`, which is precisely the statement that `x*` is a global minimum. For `f(x) = |x|` we have `0` in `[-1, 1]`, so `x = 0` is the global minimiser — established without ever differentiating at the kink. When `f` is differentiable the condition reduces to `f'(x*) = 0`, recovering the familiar test. ### Why kinks create exact zeros Consider minimising `(1/2)*(x - z)^2 + L*|x|` over `x`, where `z` is a fixed number and `L > 0`. The optimality condition is `0` in `x - z + L*df(|x|)`. For a candidate `x = 0`, the subdifferential contributes the whole interval `[-L, L]`, so the condition becomes `|z| <= L`: whenever the target `z` is small relative to `L`, the exact solution is `x = 0`, not merely something close to it. Away from 0 the condition is `x - z + L*sign(x) = 0`, giving `x = sign(z)*(|z| - L)`. The combined solution is the soft-threshold `sign(z)*max(|z| - L, 0)`. Contrast a squared penalty `(1/2)*(x - z)^2 + L*x^2`, which is differentiable everywhere. Its derivative at `x = 0` is the single number `-z`, and that vanishes only when `z` is exactly 0. The interval-valued subdifferential of the absolute value is what creates a whole range of `z` values mapping to an exact zero. This is the mathematical reason an absolute-value penalty produces exact zeros while a squared one only shrinks. ### Operating with subgradients Three practical points distinguish a candidate who has used these rather than only read about them: 1. **Any element may be chosen.** An algorithm picks one subgradient from the set — often the convention of returning a one-sided value at the kink — and the choice affects the path taken. 2. **The negative subgradient is not guaranteed to be a descent direction.** Unlike the smooth case, stepping along it can increase the objective for every step size, so the usual descent reasoning does not carry over. 3. **Fixed step sizes do not converge to the exact optimum.** Subgradient methods need diminishing steps, and their worst-case convergence is markedly slower than gradient methods on smooth problems. Where the non-smooth part is simple, methods that handle the kink exactly rather than through a subgradient step do better. ### The boundary of validity The supporting-line definition and the optimality theorem rely on convexity. For non-convex functions the object is replaced by other generalised derivatives with weaker guarantees, and quoting the convex theory without that caveat is a genuine error.