The Intermediate Value Theorem

From Connected Intervals to Intermediate Values

The previous tutorial established that every interval in \(\mathbb{R}\) is connected. This topological fact has a direct consequence for continuous functions: as the input moves through an interval, the function cannot skip a value lying between two values it attains. The Intermediate Value Theorem gives this conclusion in a familiar form and is one of the most useful ways to prove that an equation has a solution.

The theorem concerns values at two points of the domain, not necessarily values at the endpoints of the whole interval. This distinction matters when the domain is open or unbounded. If \(u\) and \(v\) belong to an interval, the segment between them is still contained in that interval, so the theorem applies to those two points even if the interval has no included endpoints.

Here, “\(y\) lies between \(f(u)\) and \(f(v)\), inclusive” means either \(f(u)\leq y\leq f(v)\) or \(f(v)\leq y\leq f(u)\). The conclusion includes the endpoint cases: if \(y=f(u)\), take \(c=u\), and if \(y=f(v)\), take \(c=v\). The substantive case is when \(y\) is strictly between the two values.

This interval formulation follows from the Intermediate-Value Property on a Connected Set established earlier in the course. An interval \(J\) is connected, and \(f\) is continuous on \(J\), so that result applies directly to \(u\), \(v\), and \(y\). Equivalently, the image of the segment between \(u\) and \(v\) is connected; connected subsets of the real line are intervals, so that image cannot omit an intermediate value.

The Domain Need Not Have Endpoints

A common misreading is to think that the theorem requires a continuous function on a closed interval \([a,b]\). That is a convenient and frequent special case, but the theorem only needs two points \(u,v\) in the domain and an interval containing them. The interval property guarantees that every point between \(u\) and \(v\) is in the domain. Continuity on that interval then gives the intermediate value.

The same reasoning works on unbounded intervals. For example, if a continuous function is defined on \((4,\infty)\), any two of its domain points determine a closed segment contained in \((4,\infty)\). The theorem is applied to those points, not to an endpoint at infinity. There is no need to assign a function value at infinity or to assume that the function has a limit there.

A Sign Change Guarantees a Root

The most common application takes the target value to be zero. A function that is negative at one point and positive at another must vanish somewhere between them. The statement remains true if one endpoint value is already zero, in which case that endpoint itself is a root.

Proof. If \(f(a)=0\), choose \(c=a\). If \(f(b)=0\), choose \(c=b\). Otherwise, \(f(a)\) and \(f(b)\) are both nonzero. The condition \(f(a)f(b)\leq0\) then forces them to have opposite signs: either \(f(a)<0<f(b)\) or \(f(b)<0<f(a)\). In either case, \(0\) lies between \(f(a)\) and \(f(b)\). By the Intermediate Value Theorem applied to the restriction of \(f\) to \([a,b]\subseteq J\), there is \(c\in[a,b]\) with \(f(c)=0\). Since \([a,b]\subseteq J\), this point belongs to the domain. \(\square\)

When the Intermediate Value Is Unique

The Intermediate Value Theorem gives existence, but it does not generally give uniqueness. For example, the function \(x\mapsto x^2\) on \([-2,2]\) takes the value \(1\) at both \(-1\) and \(1\). A useful additional hypothesis is strict monotonicity. A strictly increasing or strictly decreasing function never takes the same value at two distinct points, so an intermediate value can occur at most once.

Proof. Existence follows from the Intermediate Value Theorem applied to the restriction of \(f\) to the closed interval with endpoints \(u\) and \(v\), which lies in \(J\). For uniqueness, suppose \(c_1\) and \(c_2\) are two points between \(u\) and \(v\) with \(f(c_1)=y=f(c_2)\). If \(c_1\ne c_2\), one of them is smaller; say \(c_1<c_2\). If \(f\) is strictly increasing, then \(f(c_1)<f(c_2)\), contradicting \(f(c_1)=f(c_2)\). If \(f\) is strictly decreasing, then \(f(c_1)>f(c_2)\), giving the same contradiction. Hence \(c_1=c_2\), and the solution is unique. \(\square\)

What the Theorem Does—and Does Not—Tell You

The theorem is a qualitative existence tool. Its conclusion is that a point with a specified function value exists; it does not usually identify that point or calculate it. For equations that cannot be solved exactly, this can still be decisive. If continuity and opposite endpoint signs have been verified, one has a rigorous existence proof even when a numerical method is needed to estimate the root.

The hypotheses must be checked. Continuity cannot be omitted: define \(g(x)=-1\) for \(x<0\) and \(g(x)=1\) for \(x\geq0\) on \([-1,1]\). Then \(g(-1)=-1\) and \(g(1)=1\), but \(g\) never equals \(0\); the jump at \(0\) prevents continuity. Also, the interval condition matters. A continuous function on a disconnected domain may take values on both sides of a target while never taking that target. For instance, \(h(x)=x\) on \([-2,-1]\cup[1,2]\) takes both negative and positive values but never equals \(0\), because \(0\) is outside its domain.

When applying the theorem, a reliable sequence of checks is: identify an interval contained in the domain, verify continuity there, calculate the two function values, and confirm that the target lies between them. If uniqueness is also required, establish an additional property such as strict monotonicity; it is not part of the Intermediate Value Theorem alone.