Actually I have two related questions.

Here is the first...

Suppose $X$ is a, possibly singular, complex projective variety. Let $D$ be an effective Cartier divisor on $X$ and $x\in X$ a closed point such that the multiplicity of $D$ at $x$, usually denoted by $mult_x(D)$, is $k>0$.

Let $\mu:X'\rightarrow X$ be the blowing up at $x$, and let $E$ be the exceptional divisor.

Is it true that the order of $\mu^*(D)$ along $E$ is $k$?

I'm sure about it in the smooth case and I suppose it is true also in the singular case but I don't have a reference.

The second question is the following:

Suppose $(X,\Delta)$ is a KLT pair, and $f:Y\rightarrow X$ is a log resolution of the pair $(X,\Delta)$.

Let $E$ be an exceptional divisor on $Y$ mapping to a point on $X$ and let $a(E,X,\Delta)$ be its the discrepancy. In other words $a(E,X,\Delta)$ is the order along $E$ of the divisor $K_Y-\mu^*(K_X+\Delta)$.

I know $a(E,X,\Delta)\leq 1$ if $X$ is smooth.

It really seems to me this is also true in the singular case. Do I wrong? In any case do tou have a proof or a reference for this?

Thanks a lot!