Her husband frequently swears by divorce. Is she still married to him?

Firstly:  

The ruling on whether you are still married to him or not depends on whether divorce actually took place or not. It is well known that if a woman has been divorced three times, then she becomes irrevocably divorced from her husband and is not permissible for him until she has been married to another husband. 

What you have mentioned is that your husband frequently says, “My wife is divorced if I do such and such” then he does not do it. This is something concerning which the scholars have differed. The majority are of the view that divorce does take place if the oath is broken, i.e., if he does not do what he swore he would do.
See: al-Mughni (7/372). 

Some of the scholars – including Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) – were of the view that if a person issues a divorce in the form of an oath, but his intention was to ensure doing something or refrain from doing something, then if the oath is broken he has to offer kafaarat yameen (expiation for breaking an oath) only, and the divorce does not count as such. This is what was stated in fatwas by Shaykh Ibn Baaz and Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on them). Based on this opinion, your husband’s intention should be examined. If he intended that a divorce should take place when the condition was fulfilled, then divorce has taken place, but if he intended to prevent himself or someone else from doing something, or to compel himself or someone else to do something, then he broke the oath, then this is an oath which may be expiated by offering kafaarat yameen. See the answer to question no. 39941. 

Kafaarat yameen (expiation for breaking an oath) involves freeing a slave or feeding or clothing ten poor persons; if that is not possible then one must fast for three days. 

Secondly: 

With regard to divorce issued at a time of anger, in some cases it counts as such and in some cases it does not, depending on the type and degree of anger. For more details on that please see the answer to question no. 22034. 

Thirdly: 

The Islamic way of dealing with this problem is for your husband to understand the seriousness of uttering words of divorce and the consequences of that, as it exposes his married life to destruction. It is sufficient for him to note that his staying with his wife now is a matter concerning which the scholars differed, and the majority of them are of the view that he has divorced her three times, especially if he utters these words frequently. Perhaps he has divorced her ten times. Would a Muslim be happy for his staying with his wife to be the subject of a difference of opinion among the scholars, most of whom say that it is not permissible for you to stay with her? Verily to Allaah we belong and unto Him is our return. 

What he has to do is fear Allaah and understand the seriousness of the words that have come out of his mouth. He should not swear by anything except Allaah, and he should stop swearing by anything else. We ask Allaah to keep us and you safe and sound. 

And Allaah knows best.