Boundary Inclusion and Exclusion in Domains

Why Boundary Points Need a Check

A domain condition often describes a region using an inequality. The equality case is where a boundary may occur, but it does not always tell you whether a point on that boundary is allowed. That depends on the operation in the formula.

For an even root, a radicand of zero is allowed. For a logarithm, an argument of zero is not allowed. For a denominator, a value of zero is not allowed. These differences determine whether an edge or curve is part of the domain.

The practical rule is simple: first find the candidate boundary by replacing the inequality with equality; then substitute a point on that boundary into the original formula. The equality condition alone is not a universal instruction to include or exclude the point.

What Does Boundary Inclusion Mean?

Suppose a formula is defined on one side of a line or curve. The boundary is the edge separating that region from points outside it. A boundary point is included if it belongs to the domain, and excluded if it does not.

For example, the inequality \(x\geq 0\) describes the right half-plane together with the line \(x=0\). The inequality \(x>0\) describes the right half-plane without that line. The equality \(x=0\) identifies the candidate boundary; the inequality symbol says whether it is included.

Domain restrictions come from the formula’s operations. For the restrictions considered here, the conditions are:

These conditions apply to real-valued formulas. A zero radicand gives a real value for an even root, but a logarithm of zero and division by zero are undefined. For a formula with more than one restriction, a point belongs to the domain only if it satisfies every restriction.

Even Roots: Equality Is Allowed

The real square root \(\sqrt{u}\) is defined when \(u\geq 0\). In particular, \(\sqrt{0}=0\), so the equality case is valid. If the radicand is a function of several variables, the candidate boundary is where that radicand equals zero.

For example, a condition \(9-x^2-y^2\geq 0\) can be written as \(x^2+y^2\leq 9\). Its boundary is the circle \(x^2+y^2=9\), and the non-strict inequality includes that circle.

Logarithms: Equality Is Excluded

The real logarithm \(\ln(u)\) is defined only when \(u>0\). It is not defined at \(u=0\). Therefore, when the argument approaches zero along a boundary, the boundary points where it equals zero do not belong to the domain.

The distinction between \(u\geq 0\) and \(u>0\) is essential. Replacing the strict inequality with a non-strict one would incorrectly include points where the logarithm has no real value.

Denominators: Zero Is Excluded

A quotient is defined only where its denominator is nonzero. Thus, if a denominator is \(D(x,y)\), the condition is \(D(x,y)\neq 0\). The equation \(D(x,y)=0\) identifies points to remove from the domain.

Unlike a root condition, this is not a region that includes one side of an equality curve. The denominator’s zero set is excluded wherever it occurs. Points on either side can still be in the domain if the denominator is nonzero there.

Why the Equality Test Works

The boundary test follows directly from the definitions of the real operations involved. It is not a separate convention about geometric edges.

To justify the rule fully, let \(u\) be the value produced by a radicand, logarithm argument, or denominator at a point. The real square root exists exactly when \(u\geq 0\); therefore, if the boundary value is \(u=0\), the expression has value \(\sqrt{0}=0\). The real logarithm exists exactly when \(u>0\), so \(u=0\) fails its defining condition. A quotient with denominator \(u\) is defined exactly when \(u\neq 0\), so \(u=0\) also fails that condition. Hence the same equality value is included for the root and excluded for the other two operations.

When several restrictions occur in one formula, the domain is their intersection: a point must pass every test. An allowed equality for one operation cannot compensate for a failed condition elsewhere. For instance, a point where a square-root radicand is zero is still excluded if another part of the formula has a logarithm argument equal to zero or a denominator equal to zero.

Common Mistake: Treating Every Boundary the Same Way

A frequent error is to assume that equality always means inclusion because a boundary looks like an edge of the region. The reverse error is to assume that every equality boundary must be removed. Both mistakes ignore the operation that created the restriction.

For example, the following calculation is wrong for \(q(x,y)=\sqrt{x+y}\):

The second line improperly removes the equality case. At \((1,-1)\), the radicand is \(1+(-1)=0\), and \(q(1,-1)=\sqrt{0}=0\). The correction is to keep \(x+y\geq 0\), including its boundary line.

For a logarithm, however, changing \(>\) to \(\geq\) creates the opposite error. If \(r(x,y)=\ln(x+y)\), the condition \(x+y\geq 0\) would include points on \(x+y=0\), where the argument is zero and the logarithm is undefined. The correct condition is \(x+y>0\).

A Reliable Boundary-Checking Routine

When reading a formula, use a short sequence of checks rather than relying on the shape of a sketch:

1. Write the domain condition for each operation in the formula.
2. Use equality to locate the candidate boundary for each inequality restriction.
3. Check the equality case using the operation’s condition: nonnegative for an even-root radicand, positive for a logarithm argument, and nonzero for a denominator.
4. Substitute a specific candidate boundary point into the original formula.
5. If there are multiple restrictions, confirm that the point satisfies all of them.

A sketch can help show which side of a line or curve satisfies an inequality, but it cannot replace the equality check. A boundary may be included, excluded, or subject to different conditions at different parts of a domain when several restrictions are combined.