prove or display a desproving example:
A)
there is f and g which are defined x_{0}-\delta<x<=x_{0}+\delta for which lim_{x->x_{0}}f(x)g(x)=\infty.
if 0<g(x)<1 for x\in N_{\delta}^{*}(x_{0}) then lim_{x->x_{0}}f(x)=\infty
B)
if f monotonickly decreasing in [0,\infty) and lim_{x->\infty}f(x)=0 then f(x)>0 for all x\in[0,\infty)
C)if f differentiable in (a,b) so lim_{x->a^{+}}f(x)=lim_{x->b^{-}}f(x)=1 then there is c\in(a,b) so f'(c)=0
D)if f is integrabile in [a,b] then there is x in [a,b] so \int_{a}^{x}f(t)dt=\int_{x}^{b}f(t)dt


regarding a:
i dont know from the start how to know intuetivly whether its true or false
?