Help with a cumulative distribution function question. This is a question I want to solve:

The random variable $X$ has cdf:
$$
F_X(x) = \begin{cases}
\begin{align}
&amp;0 &amp;&amp;x &lt;0\\
&amp;0.5 + c\sin^2\left(\frac{\pi x}{2}\right) &amp;0 \leq\; &amp;x \leq 1\\
&amp;1 &amp;&amp;x&gt;1
\end{align}
\end{cases}
$$

(a) What values can $c$ assume?

(b) Plot the cdf.

(c) Find $P[X &gt; 0]$.

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.
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.
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 $0\leq x \leq 1$.