I read in my book that cdf $F(x)$ is non-decreasing function. It means it can be increasing or constant or mix of both
If i take it as constant function then from $-\infty$ to $+\infty - \delta$(delta is some small number) it is zero and at $+\infty$ it is +1. Can i define cdf in such a way? I know this is some hypothetical stupid question but wanted to know if is this possible?
The expression $+\infty-\delta$ only makes sense in context of the extended real line $\overline{\mathbb R}=\mathbb R\cup\{-\infty,\infty\}$ where by convention we would have $+\infty-\delta=+\infty$. So your question doesn't really make sense.
It is possible, though, to define an random variable e.g. $X$ such that $\mathbb P(X=+\infty)=1$. In that case the distribution function $F_X$ would satisfy $F_X(t)=0$ for all real $t$, but $F_X(+\infty)=1$.