Monotone Functions And Darboux Sums

Why Monotonicity Simplifies Darboux Sums

A Darboux sum is built by dividing an interval into subintervals and using a function's infimum or supremum on each one as a rectangle height. For a general bounded function, finding those infima and suprema may require careful analysis. For a monotone function, the endpoint values determine them directly.

This lesson focuses on that endpoint choice and on computing the resulting sums when all subintervals have equal width. The direction of monotonicity matters: for an increasing function the left endpoint gives the lower value, while for a decreasing function the right endpoint gives the lower value.

Partitions and Darboux Sums

Let \(f:[a,b]\to\mathbb{R}\) be bounded, where \(a<b\). A partition \(P\) of \([a,b]\) is a finite increasing list of points

The \(i\)-th subinterval is \([x_{i-1},x_i]\), and its width is \(\Delta x_i=x_i-x_{i-1}\). On this subinterval, let \(m_i\) be the infimum of \(f\)'s values and \(M_i\) be their supremum. The lower and upper Darboux sums are defined by

The infimum and supremum are used in the definitions whether or not they are attained. For a monotone function on a closed subinterval, however, the endpoint values do attain them: monotonicity orders every function value between the two endpoint values.

Finding the Endpoint Values

Suppose \(f\) is increasing on \([a,b]\). For any subinterval \([x_{i-1},x_i]\), every \(x\) in that subinterval satisfies \(x_{i-1}\le x\le x_i\). Therefore

Because both endpoints belong to the subinterval, the left endpoint value is its infimum and the right endpoint value is its supremum. For a decreasing function the inequalities reverse: the left endpoint value is the supremum and the right endpoint value is the infimum.

These statements remain correct when the function is constant on a subinterval: the endpoint values then agree, so the infimum and supremum are equal.

Equal-Width Partitions

An equal-width partition divides \([a,b]\) into \(n\) subintervals of the same width. For a positive integer \(n\), define

Since every subinterval has width \(\Delta x\), the endpoint rule gives compact formulas for both sums.

Each term is a function value multiplied by the width of its subinterval. The sum is not necessarily an ordinary geometric area if some function values are negative; it is the signed sum specified by the definition.

Why the Formulas Work

The formulas follow directly from the definitions, and the argument also verifies the order of the sums. Consider the increasing case first. On the \(i\)-th subinterval, monotonicity gives

The endpoint values are included in the range on that subinterval, so its infimum is \(f(x_{i-1})\), and its supremum is \(f(x_i)\). Substituting those values into the definitions of \(L\) and \(U\), and using the common width \(\Delta x\), gives the increasing-function formulas.

For a decreasing function, monotonicity instead gives \(f(x_i)\le f(x)\le f(x_{i-1})\) throughout the same subinterval. Thus its infimum is \(f(x_i)\) and its supremum is \(f(x_{i-1})\), which gives the decreasing-function formulas. In either case, every subinterval's infimum is no greater than its supremum. Since every width is positive, adding the corresponding products yields

This justification uses monotonicity to locate the extrema; it does not assume that an arbitrary bounded function attains its infimum or supremum. For monotone functions on these closed subintervals, attainment follows from the endpoint values themselves.

A Common Endpoint Mistake

A frequent error is to choose the left endpoint for the lower sum regardless of whether the function is increasing or decreasing. For \(g(x)=10-2x\) on \([1,5]\), that incorrect choice would give

as the purported lower sum. But \(g\) is decreasing, so those left endpoint values are the suprema, not the infima. The value \(20\) is the upper sum. The correction is to use the right endpoint values for the lower sum:

A reliable check is to write down the endpoint values in order before summing. If they increase from left to right, the left values are lower; if they decrease, the right values are lower. Then check that the computed lower sum does not exceed the upper sum.

Why This Endpoint Rule Is Useful

Monotonicity replaces a separate search for the infimum and supremum on every subinterval with a simple comparison of endpoint values. This makes Darboux sums practical to compute even when the function is not linear, provided its direction of monotonicity is known on the whole interval. If monotonicity holds only on part of the interval, the endpoint rule can be used only on subintervals where that monotonicity condition holds.

The method also clarifies what information a sum uses: the partition determines the widths, monotonicity determines which endpoint is lower or upper, and the function formula supplies the endpoint values. Keeping these tasks separate helps prevent both endpoint errors and arithmetic errors.