Show that the function f(x) = (E(x))/(1+x²) satisfies
Show that the function $f(x) = \dfrac{E(x)}{1+x^2}$ satisfies Dirichlet's conditions on every finite interval, where $E(x)=\lfloor x\rfloor$ denotes the integer part (floor function) of $x$.
Explanation of Method
Dirichlet's conditions require boundedness, finitely many discontinuities with finite one-sided limits, and finitely many extrema on each finite interval. Since $E(x)$ is piecewise constant, jumping only at integers, and $1/(1+x^2)$ is smooth and bounded away from zero, all four conditions reduce to counting integers in the interval.
- Bound $f(x)$ using the boundedness of $E(x)$ on any finite interval.
- Locate the discontinuities of $f$, which occur only at integers (inherited from $E(x)$), and verify one-sided limits are finite.
- Count extrema by analyzing $f$ on each interval $[n,n+1)$ where $E(x)=n$ is constant.
Step 1 · Boundedness
On a finite interval $[a,b]$, $E(x)$ takes only finitely many integer values, so there exists $M>0$ with $|E(x)|\le M$ for all $x\in[a,b]$. Since $1+x^2\ge1$ always, $$ |f(x)| = \frac{|E(x)|}{1+x^2} \le M. $$
Step 2 · Discontinuities
$E(x)$ is continuous except at integers, where it jumps: $\lim_{x\to n^-}E(x)=n-1$, $\lim_{x\to n^+}E(x)=E(n)=n$. Since $1/(1+x^2)$ is continuous everywhere, $f(x)=E(x)/(1+x^2)$ inherits discontinuities only at integers, with $$ \lim_{x\to n^-}f(x)=\frac{n-1}{1+n^2}, \qquad \lim_{x\to n^+}f(x)=f(n)=\frac{n}{1+n^2}, $$ both finite. On $[a,b]$ there are only finitely many integers.
Step 3 · Extrema
On each interval $[n,n+1)$, $E(x)=n$ is constant, so $f(x)=\dfrac n{1+x^2}$ and $$ f'(x) = \frac{-2nx}{(1+x^2)^2}. $$ For $n\ne0$, this vanishes only at $x=0$, which lies in $[n,n+1)$ only when $n=0$; for $n=0$, $f\equiv0$ has no interior extremum. So on every subinterval with $n\ne0$, $f$ is strictly monotonic, with any local extremum occurring only at the jump points (integers) themselves. On a finite interval $[a,b]$ there are finitely many integers, hence finitely many extrema.
Summary Table
| Dirichlet condition | Verification |
|---|---|
| Bounded | $|f(x)|\le M$ on any $[a,b]$ |
| Finitely many discontinuities | Only at integers in $[a,b]$ |
| Finitely many extrema | Monotonic on each $[n,n+1)$ |
| One-sided limits finite | $\frac{n-1}{1+n^2}$, $\frac{n}{1+n^2}$ at each integer $n$ |
Additional Notes
- The key structural fact driving every condition here is that $E(x)$ is piecewise constant: this alone forces finitely many jumps and (combined with the smooth, monotone-friendly factor $1/(1+x^2)$) finitely many extrema on any bounded interval.
- This problem is the first in a short sequence (321–323) checking Dirichlet's conditions directly from the definition, building toward the Fourier series machinery used from Problem 324 onward.
- Contrast with Problem 323, where a continuous, bounded function still fails Dirichlet's conditions because it has infinitely many extrema accumulating at a point.