Help with a cumulative distribution function question. This is a question I want to solve:
The random variable $X$ has cdf:
$$
F_X(x) = \begin{cases}
\begin{align}
&0 &&x <0\\
&0.5 + c\sin^2\left(\frac{\pi x}{2}\right) &0 \leq\; &x \leq 1\\
&1 &&x>1
\end{align}
\end{cases}
$$
(a) What values can $c$ assume?
(b) Plot the cdf.
(c) Find $P[X > 0]$.
I assume that $x$ is between $0$ and $1$ so
$$
0.5 + c \sin^2(\pi\times x/2) = 0
$$
then $x=1$ and
$$
c \sin^2(\pi\times 1/2) = -0.5
$$
since $c = -0.5$
Is that correct and I need help with the nother parts please.
 A: You have no reason to assume that $0.5 + c \sin^2(\pi\times x/2) = 0$.
In fact the definition of $F$ tells you that 
$$F_X(0) = 0.5 + c \sin^2(\pi\cdot 0/2) = 0.5 \neq 0,$$
so your assumption is false for at least one $x$ in the interval $0 \leq x \leq 1$.
There is also no reason to assume that $c = -0.5$.
In fact it cannot be $-0.5.$
I would suggest looking carefully at what happens around $x=1.$
It is possible for a cdf to be discontinuous, but only some kinds of discontinuity
are possible.
A: Hint: A CDF is monotonic increasing and right continuous. However, $\sin^2(x\pi/2)\in [0,1]$ 
A: The nature of a cumulative distribution function is that it must be càdlàg, monotonically non-decreasing, and take values between 0 and 1 (inclusive). It does not have to be either continuous or discrete, but may be both.


*

*Part 1: Being that the distribution may not take values greater than one, and must be non-decreasing, what can the maximum of $c$ be if we know it is equal to $0.5 + c\sin^2\left(\frac{\pi x}{2}\right)$? What is $\sin\left(\frac{\pi}{2}\right)$? Similarly, what is the minimum it can be, remembering that it must be non-decreasing?

*Part 2: Being that $c$ is a constant, there are really only two kinds of shapes it can have, so once you figure out part 1, this isn't so difficult

*Part 3: Make sure you recognize those inequalities which are strict, and those that aren't, and what the difference must mean.

A: I think the CDF given in the question is wrong as in your problem it is neither a continuous CDF nor a discrete one as LHL (left hand limit) approaching 0 and RHL (right hand limit) approaching 1 do not converge to the same value. Which in this case is not true for F(x) at $0\leq x \leq 1$.
