Familiarize yourself with limits. Limits describe the behavior of a function at a certain point. You can take the limit of a function by looking at the behavior of the function as it approaches that number or by directly plugging the number into the function.
Decide if you want to find continuity at a single point or for an entire function.
Graph the function or take the limit as the function goes to infinity to determine continuity for a function. If there is no break in the graph or if the limit exists, then the function is continuous. The limit does not exist where the function is divided by zero or where it "equals" negative or positive infinity.
Take the limit as the function approaches a point C from the left and the limit as the function approaches point C from the right to determine continuity at point C. If the limit from the left and the limit from the right exist and are the same, then the function is continuous.