Determine all functions satisfying $f(x)=\frac{n}{2}\int^{x+\frac{1}{n}}_{x-\frac{1}{n}} f(t)\,dt$

Question

Question: Determine all functions $f:\mathbb{R}\to \mathbb{R}$ that satisfy the following two properties.

1. The Reimann integral $\displaystyle \int^b_a f(t)\,dt$ exists for all real numbers.
2. For every real number $x$ and every integer $n\geq 1$, we have $\displaystyle f(x)=\frac{n}{2}\int^{x+\frac{1}{n}}_{x-\frac{1}{n}} f(t)\,dt$.

My attempt: I differentiated both sides to get $f'(x)=\dfrac{n}{2} \left(f\left(x+\dfrac{1}{n}\right)-f\left(x-\dfrac{1}{n}\right)\right)$ which is the same as the mean value theorem form. Thus, the gradient of any secant drawn between points with $x$-coordinates $a+\frac{1}{n}$ and $a-\frac{1}{n}$ is equal to the gradient at $a$ for any $a\in\mathbb{R}$. Geometrically, this means we can only have a continuous linear function (of the form $f(x)=mx+b$ for $m,b\in\mathbb{R}$) and it is easy to verify that this is true.

My questions:

1. How can we prove this more rigorously using calculus? My attempt was simply an intuitive geometric interpretation of the mean value theorem and does not show that this is all the possible functions.

2. Where does the "integer $n\geq 1$" part come into play? I am unsure of why there is a restriction for $n$ to be an integer. Can it not be rational or irrational $n\geq 1$?

Answer 1

We start off with the hint that @WhatsUp gave. Note that for all $n$ we have

$$\frac{1}{n}\left[f\left(x-\frac{1}{2n}\right)+f\left(x+\frac{1}{2n}\right)\right]=\int_{x-\frac{1}{n}}^{x}f(t)dt+\int_{x}^{x+\frac{1}{n}}f(t)dt=\int_{x-\frac{1}{n}}^{x+\frac{1}{n}}f(t)dt=\frac{2}{n}f(x)$$

$$\Rightarrow f(x)=\frac{f\left(x-\frac{1}{2n}\right)+f\left(x+\frac{1}{2n}\right)}{2}$$

Now, fix $f(0)=a$ and $f(1)=b$. The details are slightly tedious, but we are able to show by induction that the above equation as well as the knowledge of $f(0)$ and $f(1)$ imply

$$f(x)=(b-a)x+a$$

for all

$$x\in S=\left\{\frac{k}{2^r}:k\in\mathbb{Z}\text{ and }r\in\mathbb{N}\right\}$$

(see below for induction proof). Going forward, define $g(x)=f(x)-(b-a)x-a$. Thus, $g(x)$ is $0$ on the set $S$ and $g(x)$ satisfies

$$g(x)=f(x)-(b-a)x-a=\frac{n}{2}\int_{x-\frac{1}{n}}^{x+\frac{1}{n}}f(t)dt-(b-a)x-a$$

$$=\frac{n}{2}\int_{x-\frac{1}{n}}^{x+\frac{1}{n}}[g(t)+(b-a)t+a]dt-(b-a)x-a=\frac{n}{2}\int_{x-\frac{1}{n}}^{x+\frac{1}{n}}g(t)dt$$

the same integral relationship as $f(x)$. Now, let $y$ be any real not already in $S$ and let $y_m$ be a sequence of rational numbers of the form $\frac{k}{2^r}$ (for integral $k$ and $r$) that approaches $y$. Further, define $z_m=y-y_m$ (note that $z_m$ approaches $0$). Then

$$g(y)=\frac{1}{2}\int_{y-1}^{y+1}g(t)dt$$

$$=\frac{1}{2}\int_{y_m-1}^{y_m+1}g(t)dt+\frac{1}{2}\int_{y-1}^{y-1-z_m}g(t)dt+\frac{1}{2}\int_{y+1-z_m}^{y+1}g(t)dt$$

$$=g(y_m)+\frac{1}{2}\int_{y-1}^{y-1-z_m}g(t)dt+\frac{1}{2}\int_{y+1-z_m}^{y+1}g(t)dt$$

$$=\frac{1}{2}\int_{y-1}^{y-1-z_m}g(t)dt+\frac{1}{2}\int_{y+1-z_m}^{y+1}g(t)dt$$

But by the first assumption, we know $g(t)$ is Riemann integrable. This implies that for $m$ such that $|z_m|\leq 1$ we have

$$\left|\frac{1}{2}\int_{y-1}^{y-1-z_m}g(t)dt+\frac{1}{2}\int_{y+1-z_m}^{y+1}g(t)dt\right|\leq z_m \sup_{t\in (y-4,y+4)}g(t)$$

Again, since $g(x)$ is Riemann integrable, the supremum of $g(x)$ over the interval $(y-4,y+4)$ must exist (see here for a good description regarding this). This implies

$$0\leq |g(y)|\leq z_m \sup_{t\in (y-4,y+4)}g(t)$$

Since this is true for all but a finite number of $m$ and $z_m$ approaches $0$, we conclude

$$0\leq |g(y)|\leq \lim_{m\to\infty} z_m \sup_{t\in (y-4,y+4)}g(t)=0$$

Thus, $g(y)=0$ and we are done.

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Induction proof: We will proceed with a double induction on the tuples $(k,r)$ for $k\in\mathbb{Z}$ and $r\in\mathbb{N}$.

First, note that at $n=1$ we have

$$f\left(\frac{1}{2}\right)=\frac{f(0)+f(1)}{2}=\frac{b+a}{2}=\frac{b-a}{2}+a$$

as desired. Thus, the proposition is true for $(0,1),(1,1),(2,1)$. Assume it is true for $(k,1)$ such that $k\in \{0,1,...,m\}$. Then

$$f\left(\frac{m}{2}\right)=\frac{f\left(\frac{m+1}{2}\right)+f\left(\frac{m-1}{2}\right)}{2}$$

Then solving for our unknown gives us

$$f\left(\frac{m+1}{2}\right)=2f\left(\frac{m}{2}\right)-f\left(\frac{m-1}{2}\right)$$

$$=(b-a)m+2a-(b-a)\frac{m-1}{2}-a=(b-a)\left(\frac{m+1}{2}\right)+a$$

as desired. A similar construction works for negative $k$. Now, assume the proposition is true for all $(k,r)$ where $k\in\mathbb{Z}$ and $r\in\{1,2,...,m\}$. If $k$ is even, then the proposition is proved since for $k=2s$ we have

$$\frac{k}{2^{m+1}}=\frac{2s}{2^{m+1}}=\frac{s}{2^m}$$

If not, then for $n=2^m$ we have

$$f\left(\frac{k}{2^{m+1}}\right)=\frac{f\left(\frac{k}{2^{m+1}}+\frac{1}{2^{m+1}}\right)+f\left(\frac{k}{2^{m+1}}-\frac{1}{2^{m+1}}\right)}{2}$$

$$=\frac{f\left(\frac{k+1}{2}\cdot\frac{1}{2^{m}}\right)+f\left(\frac{k-1}{2}\cdot\frac{1}{2^{m}}\right)}{2}$$

Note that both $\frac{k\pm 1}{2}$ are integral since $k$ is odd. Then by our inductive assumption we have

$$=\frac{(b-a)}{2}\left[\frac{k+1}{2}\cdot \frac{1}{2^m}+\frac{k-1}{2}\cdot \frac{1}{2^m}\right]+\frac{1}{2}(a+a)=(b-a)\frac{k}{2^{m+1}}+a$$

as desired.