Testing Whether a Relation Defines y Locally

Why Check Whether an Equation Defines One Local Branch?

An equation \(F(x,y)=0\) describes a set of points in the plane. Near a particular point on that set, we may want to treat \(y\) as a function of \(x\). That requires more than being able to differentiate the equation formally: there must actually be one suitable \(y\)-value for each nearby \(x\), and those values must vary differentiably.

The key test is the partial derivative with respect to \(y\). When it is nonzero, and \(F\) has continuous first partial derivatives near the point, the implicit function theorem guarantees a unique differentiable local branch. Both parts of this condition matter.

Here, \(F_x\) and \(F_y\) are the partial derivatives of \(F\) with respect to its first and second variables. A function with continuous first partial derivatives is often called \(C^1\). The theorem requires \(F\) to be \(C^1\) on an open region containing the point, not just to have a nonzero value of \(F_y\) at the point.

What “Locally” and “Uniquely” Mean

The conclusion concerns a small neighborhood of \((a,b)\), not necessarily the entire curve. There is an interval of \(x\)-values around \(a\) and a range of \(y\)-values around \(b\) such that each \(x\) in that interval corresponds to exactly one solution \(y\) in that range. Those solutions make up \(y=g(x)\).

This local qualification matters. A curve could have several \(y\)-values for the same \(x\) far from \((a,b)\) without affecting the theorem’s conclusion near \((a,b)\). The theorem also does not claim that every equation defines a function everywhere, or that the equation can always be rearranged into a simple explicit formula.

Why a Nonzero \(F_y\) Matters

If \(F_y(a,b)\ne 0\), continuity of \(F_y\) means it keeps the same sign in a sufficiently small rectangle around \((a,b)\). For each fixed \(x\) in that rectangle, \(F(x,y)\) therefore changes strictly in one direction as \(y\) changes. It cannot take the value zero at two different \(y\)-values there. This is the source of local uniqueness.

Existence is also needed: for nearby \(x\)-values, the function must actually reach zero between two nearby \(y\)-values. The continuity of \(F\), together with \(F(a,b)=0\) and the consistent direction of change in \(y\), ensures that the values at suitable upper and lower \(y\)-boundaries have opposite signs. The intermediate value theorem then supplies a root between them.

Deriving the Branch’s Derivative

Once the theorem guarantees that \(y=g(x)\), the equation \(F(x,g(x))=0\) holds for all \(x\) in a small interval. Differentiate this identity using the chain rule. The first input of \(F\) changes with derivative \(1\), while the second input \(g(x)\) changes with derivative \(g'(x)\):

Because \(F_y\) remains nonzero in a sufficiently small neighborhood, we can solve for \(g'(x)\):

This derivation explains the formula, but it depends on the theorem: we first need an actual differentiable function \(g\) to differentiate. The algebraic formula alone does not establish that the equation defines such a function.

Worked Examples

A Justification of Local Existence, Uniqueness, and Differentiability

The main ideas behind the theorem can be made precise in the two-variable case. Since \(F_y(a,b)\ne0\) and \(F_y\) is continuous, choose a small rectangle around \((a,b)\) where \(F_y\) keeps its sign. At \(x=a\), the mean value theorem in the \(y\)-direction then shows that \(F(a,y)\) has opposite signs at \(y=b-s\) and \(y=b+s\), for a sufficiently small positive \(s\). By continuity, those opposite signs persist when \(x\) is sufficiently close to \(a\).

These steps account for why the theorem provides a unique \(C^1\) branch, rather than merely a formal slope calculation. The nonzero partial derivative controls how the equation changes in the \(y\)-direction; the continuity assumptions ensure that this control persists near the point.

Common Pitfall: Checking \(F_y\) but Ignoring \(C^1\)

A nonzero value of \(F_y\) at one point does not by itself guarantee a continuous branch. For a concrete example, define \(q(0)=0\) and \(q(x)=1\) for \(x\ne0\), and set \(F(x,y)=y+q(x)\). At \((0,0)\), \(F(0,0)=0\), and the partial derivative with respect to \(y\) is \(F_y=1\). But \(F\) is not continuous near \((0,0)\), so the theorem’s hypotheses are not satisfied.

The equation forces \(y=0\) when \(x=0\), but \(y=-1\) for every \(x\ne0\). Those values cannot be part of a continuous function through \((0,0)\). Thus it would be incorrect to conclude that a differentiable local branch exists just from \(F_y(0,0)\ne0\). Always check all the hypotheses, including continuous first partial derivatives on an open region around the point.

Why This Test Is Useful

Implicit differentiation gives information about how a branch would slope if it exists. The implicit function theorem connects that calculation to the geometry of the relation: it tells us when there really is one smooth local graph whose derivative the formula describes. This is especially useful when solving explicitly for \(y\) would be difficult or would obscure which branch passes through the point.

When applying the test, check the conditions in order: verify that the point satisfies the equation; verify that \(F\) is \(C^1\) on an open region containing it; then evaluate \(F_y\) at the point. If all conditions hold and that value is nonzero, a unique differentiable local branch exists. If \(F_y=0\), do not infer either that a branch exists or that one fails to exist—the test simply does not settle the question.