Questions
I am looking for some counterexamples for properties that of functions that are commonly mistaken for being true in cases where they do not hold in general. The specific cases I am thinking of are the following:
Give an example of a regulated function $f$ (that can be approximated by a sequence of step functions $\phi_n$) on $[0, 1]$ such that $f (x) ≥ 0$ for all $x ∈ [0, 1]$, $\int_0^1f(x)dx = 0$, and such that $f(x_0) > 0$ for at least one $x_0 ∈ [0,1]$.
Is there an example which is not a step function?
Attempt
My initial idea for the first part of the problem is to define a step function on a suitable partition $P=\{p_0,p_1,...p_n\}$ as $\psi(x)=0$ for any $x \in (p_{i-1},p_i)$, and $\psi(p_i)=1$ (at the endpoint of each subinterval).
My initial idea with this problem the second part of the problem is to define a sequence of step functions $\phi_n=1$ on the interval $\big{[}1,\frac{1}n\big{]}$ and $0$ otherwise. Then as $n \rightarrow \infty$, we say that $\psi \rightarrow f$ and so this provides an example that is not a step function.
Are these examples fine, or do they violate the definitions in some way that I am not seeing? The main concern I have is with the second part. I know that the integral/area should be converging to $0$, however, I'm not convinced that it is equal to $0$ in the same way that it was with the original step function (since this was a set of points rather than a thin strip).
The Thomae Function is a regulated function with $0$ integral. This is a regulated function and is therefore Riemann integrable. The values of the function are greater than or equal to $0$ (strictly positive where $x$ is rational).
See below, a plot of values on the interval $(0,1)$ (with the highest point in the middle at $(\frac{1}2,\frac{1}2)$ - taken from the Wikipedia page linked above.