Let $(X,d)$ be  a  compact metric space. Assume  that $f:X\to \mathbb{R}$ is  a positive continuous  function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for  every $\epsilon>0$ there exists a finite open cover  $D_i,\;i=1,2,\ldots n $ consists of  open discs $D_i$ such that $$\sum_{i=1}^n (diam D_i)^{f(t_i)} \leq \epsilon,\; \forall t_i\in D_i$$

This  situation is  denoted  by $\mathcal{H}^f(X)=0$

We  define the  generalized Hausdorff  dimension of $X$ with $$dim(X)=\inf \{f\mid \mathcal{H}^f(X)=0\}$$

So the dimension is no longer a number but is  a  lower semi continuous.

>For  what kind of metric space this dimension is  a continuous function? For  what kind of  metric spaces this  dimension is  a  constant function(equal to its  Hausdorff dimension)?