I presume the question is to find the expected value of the random variable with the given cumulative distribution function. The distribution function is a staircase function with finite jumps at $2$ points, namely $0,1$, so the random variable $X$ must be discrete and takes only two values $0,1$ (Bernoulli). So$$P(X=0)=F(0)-F(0^-)=1/2\\P(X=1)=F(1)-F(1^-)=1/2\\\Bbb E[X]=0\times1/2+1\times1/2=1/2$$
The distribution function is still discontinuous at $1$ but non-constant in $[0,1)$, so it seems like your random variable is a mix of a continuous and discrete random variable.
The relation $P(X=1)=F(1)-F(1^-)=1/2$ remains intact.
In $[0,1)$, the PDF is given by $F'(x)=1/2$.
So the required expectation is $\int_0^1\frac x2dx+1\times1/2=3/4$.