Why Endpoint Derivatives Need Special Care
For an accumulation function, the derivative at an interior point is often connected to the value of the integrand there. At an endpoint, however, there is only one direction from which the domain can be approached. The relevant question is whether the average value of the integrand over a short interval next to that endpoint settles toward a finite number.
A one-sided limit of the integrand gives a direct way to answer that question. It also highlights an important distinction: the value assigned to the integrand at the endpoint need not be the value that determines the accumulation function’s one-sided derivative.
Accumulation Functions and Endpoint Derivatives
Let \(f\) be integrable on \([a,b]\), and define the accumulation function
At the left endpoint \(a\), the domain allows us to approach only from the right. The right-hand derivative is the limit of the corresponding difference quotient as \(h\) approaches zero through positive values. At the right endpoint \(b\), the left-hand derivative uses negative \(h\).
when these limits exist as finite real numbers. These are one-sided derivatives, not two-sided derivatives: the quotient is considered only for points in the domain.
The Difference Quotient Is an Average
Since \(F(a)=0\), the right-hand difference quotient at \(a\) can be rewritten using the definition of \(F\):
This is the average value of \(f\) over \([a,a+h]\). At the right endpoint, additivity of definite integrals gives the corresponding expression:
Although \(h\) is negative in the second expression, the integral is also oriented from \(b\) toward \(b+h\). The quotient still represents an average over the short interval between those points.
Why a Finite One-Sided Limit Gives the Derivative
Suppose the integrand approaches a finite number \(L\) from inside the interval next to an endpoint. On a sufficiently short interval, all its values are close to \(L\). Their average must then be close to \(L\) as well. This is the key justification for using the integrand’s one-sided limit.
Here is a proof at \(a\). Given any \(\varepsilon>0\), the definition of the right-hand limit provides a \(\delta>0\) such that \(0<t-a<\delta\) implies \(|f(t)-L|<\varepsilon\). For \(0<h<\delta\), the integral average therefore satisfies:
Thus the difference quotient can be made arbitrarily close to \(L\) by taking \(h\) sufficiently small and positive. That is exactly the definition of \(F'_+(a)=L\). For \(b\), the same estimate applies to the interval from \(b+h\) to \(b\), with \(h<0\); dividing the oriented integral by \(h\) again gives an average. This proves the left-endpoint result too.
Worked Examples at Endpoints
Worked Example: A Right-Hand Limit Determines the Derivative
Define \(f\) on \([0,1]\) by \(f(0)=9\) and \(f(t)=2+3\sqrt{t}\) for \(t>0\). Let \(F(x)=\int_0^x f(t)\,dt\). Find \(F'_+(0)\).
For \(t>0\), the integrand approaches \(2\) as \(t\to0^+\). The value \(f(0)=9\) does not change the integral: changing an integrable function at one point does not change its definite integral. Directly, for \(x\ge0\),
Since \(F(0)=0\), for \(h>0\) the difference quotient is
Taking \(h\to0^+\) gives \(F'_+(0)=2\), agreeing with \(\lim_{t\to0^+}f(t)=2\), not with the assigned endpoint value \(f(0)=9\). Check: the quotient \(2+2\sqrt h\) tends to \(2\), so the proposed derivative is verified.
Worked Example: A Left-Hand Limit at the Right Endpoint
On \([0,2]\), let \(f(t)=4t\) for \(t<2\) and \(f(2)=-7\). Define \(F(x)=\int_0^x f(t)\,dt\). Find \(F'_-(2)\).
As \(t\to2^-\), \(f(t)=4t\to8\). For \(0\le x\le2\), the isolated value at \(t=2\) does not affect the integral, so
For \(h<0\) close to zero, \(2+h\) remains in the domain. The endpoint difference quotient is
As \(h\to0^-\), this quotient tends to \(8\). Therefore \(F'_-(2)=8\), matching the left-hand limit of \(f\), even though \(f(2)=-7\). Substitution into the quotient confirms the result: its limit is \(8\).
A Limit Is Sufficient, but Not Necessary
The theorem gives a reliable sufficient condition: a finite one-sided limit of the integrand guarantees the corresponding finite one-sided derivative. It does not say that the integrand must have a one-sided limit whenever the derivative exists. The derivative depends on averages, and averages can settle toward one number even when individual integrand values keep oscillating.
Worked Example: An Oscillating Integrand with a Derivative
Let \(f(0)=0\) and \(f(t)=\sin(1/t)\) for \(0<t\le1\). Define \(F(h)=\int_0^h f(t)\,dt\). Determine whether \(F'_+(0)\) exists.
The integrand has no right-hand limit at zero. For example, along \(t_n=1/(\pi/2+2\pi n)\), its values are \(1\), while along \(s_n=1/(3\pi/2+2\pi n)\), its values are \(-1\). Both sequences approach zero.
To examine the derivative, substitute \(u=1/t\) in the integral. The improper-integral expression is valid because the original integrand is bounded and integrable on \([0,h]\):
Write \(A=1/h\). Integration by parts on \([A,R]\), followed by \(R\to\infty\), gives
Consequently,
Since \(F(0)=0\), it follows that
Therefore \(F'_+(0)=0\), despite the lack of a right-hand limit for \(f\). The bound verifies the derivative directly.
Using the Test Carefully
When the integrand has a finite one-sided limit, compare that limit with the corresponding endpoint derivative. If the limit is \(L\), the derivative is \(L\). If the integrand’s assigned value at the endpoint differs, that alone does not alter the conclusion.
If the integrand has no finite one-sided limit, the theorem does not settle the question. The derivative might fail to exist, or, as in the oscillating example, it might still exist. In that case, return to the difference quotient and analyze the average directly. A one-sided infinite limit also cannot give a finite derivative: for instance, if \(f(t)\to+\infty\) at \(a\), then for every \(M>0\), \(f(t)>M\) sufficiently near \(a\), so its short-interval average exceeds \(M\). The quotient cannot converge to a finite real number.
Check Your Understanding
Use the endpoint difference quotient, and distinguish a limit of the integrand from its value at the endpoint.
- If \(f(t)=5-2t\) for \(t>0\), with any assigned value at \(t=0\), what is the right-hand derivative at zero of \(F(x)=\int_0^x f(t)\,dt\)?
- If \(f(t)=3t^2\) for \(t<1\) and \(f(1)=0\), what is the left-hand derivative at \(1\) of \(F(x)=\int_0^x f(t)\,dt\)?
- State the average-value expression for the right-hand difference quotient of \(F(x)=\int_a^x f(t)\,dt\) at \(a\).
- Does a finite right-hand derivative of an accumulation function require the integrand to have a right-hand limit? Explain using the oscillating example.
- If \(f(t)\to+\infty\) as \(t\to a^+\), why can the right-hand difference quotient not approach a finite number?
