I have the following problem: Consider a random variable $x\sim F$, where $F$ is some continous probability distribution with support $[\underline{x},\bar{x}]$. Also assume there is some threshold $y\in [\underline{x},E[x]]$. Then the expected value of $x$ conditional on $x\leq y$ is $$E[x\,\big|~x\leq y]=\frac{\int_{\underline{x}}^y x\,f(x)dx}{\int_{\underline{x}}^{y} f(x)dx}~.$$
I need to establish whether and when $\frac{\partial}{\partial y}E[x\,|~x\leq y]<1$.
My intuition is that it must always be lower than one, since increasing the upper limit of the conditional support has two effects. It adds a very small amount to the numerator of the fraction above (small because continous distribution). And it increases the normalizing value in the denominator of the fraction.
Is my intuition right? And how can I prove it? Any ideas would help me a lot, as I have pondered this problem for a couple of months now.
Thanks in advance!