I've heard some professors say before that "You can switch from pure math to applied math, but not the other way". If I remember correctly, it was Munkres that said this.
There's nothing wrong with studying pure mathematics. Even with a pure math focus, you will have the ability to look at any other piece of mathematics, learn the definitions and gain a bit of intuition for it, and from there you should be able to grasp the material without much effort; mathematics isn't simply the study of a bunch of material to memorize theorems or solve problems. It teaches a way of looking at problems, about how to abstract things and how to find the relationship between two seemingly disparate topics.
You can certainly find careers that are not in academia even if you have a pure math focus. As an example, at Microsoft there is Microsoft Research New England and the Theory Group (though admittedly this borders on theoretical CS). I've known people that were pure math majors and ended doing Law School (their application was based on the idea that "Every law school needs a mathematician!"). The NSA and like organizations are also friendly towards pure mathematicians.
In the same regard, you shouldn't immediately blast applied math as being a sell-out (if I'm understanding what you mean well). Take an applied math course and see how you like it.
However, perhaps most importantly is the fact that you find pure math more interesting and invigorating, then you should almost definitely go with that.
As a current student myself, I can't say much about how taking certain classes has guided my career, but I can at least say that I'm happier doing pure math than not.
A class that has been incredibly helpful to me in the study of mathematics has been topology, which provides a new and important way at looking at different fields of mathematics, and makes precise a lot of intuitions gained over the years in conventional math courses. Other classes that would be helpful include proof based classes such as a discrete math course or (at least this is what it is at MIT) an analysis course. Most importantly, however, take what you think you'll enjoy. There is a lot of mathematics out there, and you'll almost definitely be able to find a niche in what you enjoy!