Some Problems in the book ‘A course in Probability Theory’, K.L. Chung Q1.(page10.6) A point $x$ is said to belong to the support of the d.f.(distribution funtion) $F$  iff for every $\epsilon>0$ we have $F(x+\epsilon)-F(x-\epsilon)>0$. The set of all such $x$ is called the support of $F$. Show that ezch point of jump belongs to the support, and that each isolated point of the support is a point of jump. Give an example of a discrete d.f. whose support is the whole line.
I cannot show the second claim: each isolated point of the support is a point of jump.
Q2.(page12.3) If the support of a d.f. is of measure zero then $F$ is singular. The converse is false.
I just cannot construct a proper conterexample

(1) Chung's example [Example 2. on the $3^d$ page]: Take an ordering of the rationals and assign the geometric distribution to the ordered set of rationals. Let $b_1=\frac{1}{2}, b_2=\frac{1}{4},...b_n=\frac{1}{2^{n}},...$, where $b_n$ is the mass assigned to the $n^{th}$ rational, $a_n$. All reals belong to the support of the corresponding $F$ because for any $\epsilon>0$ there are rationals in the interval $(x-\epsilon, x+\epsilon)$, that is $F$ jumps infinitely many times within that interval. So the support of this $F$ is the whole real line.
(2) Chung's definition [$11^{th}\ page]$ ]: A function is singular iff it is not identically zero and its derivative exits and is zero a.e.
Chung' theorem [$1.3.1 (a)$, $12^{th}$] page: If a function is increasing, tends to zero if $x$ tends to the $-\infty$, and is bounded then its derivative exists a.e. (${+\infty}$ as a value of the derivative is allowed.) 
Chung's example above describes such a function: it is monotonously increasing and bounded, and tends to zero  if $x$ goes to the $-\infty$, that is its derivative exists a.e. The derivative is zero at irrational points (this I cannot prove but having read Chung's first chapter I cannot get to any other conclusion.) and is ${+\infty}$ at the rationals. That is, we have a singular function whose support is the whole real line even if it is singular.
$$Edited$$
The definition of singularity is given above. As far as discreteness: Chung says on the $10^{th}$ page "A plausible definition of a discrete d.f. may be given us: >>It is a d.f. that has jumps and is constant between the jumps<<." This means that the derivative is a.e. zero since there are only countably many jumps. That is, such a d.f. is singular.
