Let $(X,\tau)$ be a Tychonoff Topological space.

For each $x\in X$ consider an arbitrary positive real number $\epsilon_x>0$. Is There a continuous real valued function $f:X\rightarrow \mathbb{R}$ with the following property:

$$\forall x \in X $$

$$0< f(x) < \epsilon_x$$

From the following comment of Edgar, I have Known that the following Question is the main Purpose of posing this Question, which I didn't notice to write it.

Q.For which properties on $(X,\tau)$, we have the above Property? (one of the properties for which the above condition is true is that $(X, \tau)$ be discrete)

Statement: Is the only property "discreteness" of $X$ ?