Presumably you are using matlab's 'roots' function, which for fourth degree polynomials will give four roots, so your task is to select out of those four, as your parameter changes, that root which is regarded as a "physical" root in your particular application.
There does exist a specific formula for solving fourth degree polynomial equations, but it is quite complicated. I would suggest instead making use of that same discontinuity you have observed in conjunction with 'roots'.
Provide the changing parameter in very small and equal incremental changes, and having started with a particular root, always select the root among the four which is closest in absolute difference to the last one chosen. That should provide some kind of continuity. It would be roughly analogous to the use of matlab's 'unwrap' function.
Of course there is the possible problem of arriving at duplicate roots for a particular parameter value. You might have trouble deciding which one of these to follow after that point. Discontinuity can also be tested at the first derivative level if you use sufficiently small and accurate parameter increments, and that might be used to resolve such a dilemma.
The statements I have made above are necessarily vague since I have no way of knowing the nature of your coefficients' dependence on your parameter and its effect on the roots.
