Can PMF have value greater than 1? Can a Probability Mass Function can have value greater than 1 like we have in Continuous functions(pdf) and why? 
Please give 1 example if the answer is yes.
And If the answer is no then why PDF can have value more then 1. As the basic definition of both function is same. F(x):Pr(R=x).
 A: Your question has 2 parts


*

*Probability Mass Function: It have discrete values and we count only those values for probability. So F(x):Pr(R=x) is only the probabilty which is always less than 1

*Probability Density Distribution: Here we don't have discrete values and if we consider a point Pr(of single point)=0.
Hence we consider area during continous distribution counting.
Now since we don't know the exact points(x) for probability distribution there can be cases when probability can be concentrated on a small points rather than the asked points(x). 
And we also know that area under such PDF curve should be 1. 
Hence if x is less than 1 then in order to keep area sum 1 value of F(x) should be greater than 1
A: No, a probability mass function cannot have a value above 1. Quite simply, all the values of the probability mass function must sum to 1. Also, they must be non-negative. From here it follows that, if one of the values exceeded 1, the whole sum would exceed 1. And that is not allowed.
A: By PMF, I assume you mean what is usually called the pdf.  For a continuous distribution the answer is yes.  What has to be true is that:
$$\int_{-\infty}^{\infty}p(x) dx = 1   $$
Imagine a normal distribution, centered at zero (so a mean of zero), and a timy standard deviation (say, .01).  Almost all of the points on that curve that will contribute much to that integral will be between -0.1 and 0.1.  So, thinking geometrically and approximating with a rectangle, the height is going to be at least something like 5 (it will actually be a lot bigger, because most of the integral will come from points between -.04 and .04).
The pdf if a non-continuous variable can never be more than 1, since (1) the sum of them must all add to 1 and (2) they must all be non-negative.  The largest it can be is if there is only one possible outcome, which then has P(x) = 1. 
By the way, the height (greater than 1) has no real meaning in itself, only as part of the integral.   It's really tempting for people, when moving from discrete to continuous, to plot out a normal curve with mean = 0 and stdev = 1.  If I recall, the high point has a value of around 0.4.  But that doesn't mean you will get zero with probability 0.4; in fact, the probability of getting exactly zero (or any other specific number) is zero.  But if you change the bounds of the integral to be $x_1$ and $x_2$, you get the probability of getting a number between $x_1$ and $x_2$.
