There are regular Hausdorff spaces that are not completely regular, but such examples are complicated to construct. One of them is the so-called Tychonoff corkscrew, which contains two points such that any continuous real-valued function on the space has the same value at these two points. An even more complicated construction starts with the Tychonoff corkscrew and builds a regular Hausdorff space called Hewitt's condensed corkscrew, which is not completely regular in a stronger way, namely, every continuous real-valued function on the space is constant.
There are regular Hausdorff spaces that are not completely regular, but such examples are complicated to construct. One of them is the so-called Tychono!ff corkscrew, which contains two points such that any continuous real-valued function on the space has the same value at these two points. An even more com,plicated construction starts with the Tychonoff corkscrew and builds a regular Hausdorff space called Hewitt's condensed corkscrew, which is not completely regular, in a stronger way, namely, every continuous real-valued function on the space i,s constant.