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
 A: (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.
A: The edited solution posted by zoli is only valid when the discrete d.f. is "discrete in the Euclidian topology" - that is, if all jump points are isolated. Question 5 on page 10 asks the reader to describe why that "plausible definition" is incorrect. It is not correct because the jump points may be dense in ℝ, in which case any two jump points will have a jump point in between, so "constant between jumps" doesn't make sense.
If the jump points are dense, then it is not immediate that a discrete d.f. is singular. How do we know that those jumps approaching each number in ℝ do not induce a positive derivative?
To see that it is singular, take the following facts from the text.
If $F$ is a d.f. then $F$ may be written as $$F = F_{ac} + F_s$$ where $F_s$ is singular, and $$0 <= F_{ac}(x) = \int_{-\infty}^{x}F'(t)dt <= F(x)$$
Then, let $F$ be a discrete d.f. and fix $\epsilon > 0$. Let $\{ a_i \}$ be an enumeration of all jump sizes. Since $F$ is discrete, $$ \sum_{i = 1}^{\infty}a_i = 1 $$ and there is some $N$ such that $$ \sum_{i = 1}^{N}a_i > 1 - \epsilon $$
Let $J = \sum_{i = 1}^{N} a_i\delta_i$ where $\delta_i$ is the point-mass function defined in the text for constructing discrete d.f.'s. Since $N$ is finite, $J$ is, in fact, "discrete in the Euclidian topology", with constant values between isolated jump points, so $J' = 0$ almost everywhere.
Now, let $G = F - J$, essentially F without the jump points which bring it within $\epsilon$ of 1, so $G(x) < \epsilon$ for all $x$ and: $$F_{ac}(x) = \int_{-\infty}^{x}F'(t)dt = \int_{-\infty}^{x}G'(t)dt + \int_{-\infty}^{x}J'(t)dt = \int_{-\infty}^{x}G'(t)dt + 0 <= G(x) < \epsilon$$ $\epsilon$ and $x$ are arbitrary and $F_{ac} >= 0$, so $F_{ac} = 0$.
From the above, $F = F_{ac} + F_s = 0 + F_s$, which is singular, which proves Q6.
Finally, the counter-example to the converse in Q2 is a discrete d.f. which is dense in some subset, $S$ of ℝ where $m(S) > 0$.
