Continuity is a fundamental concept in calculus that describes the smoothness of a function. A function f(x) is said to be continuous at a point x = a if three conditions are satisfied: (1) f(a) is defined, (2) the limit of f(x) as x approaches a exists, and (3) the limit of f(x) as x approaches a is equal to f(a). In simpler terms, a function is continuous if its graph can be drawn without lifting the pen from the paper.

On the other hand, a function is said to be discontinuous at a point x = a if one or more of the three continuity conditions are not satisfied. Discontinuities can manifest in different ways, such as removable, jump, and infinite discontinuities.

For example, consider the function f(x) = 1/x. This function is continuous everywhere except at x = 0, where it has a removable discontinuity. At x = 0, f(x) approaches positive infinity as x approaches 0 from the positive side, and negative infinity as x approaches 0 from the negative side. However, f(0) is not defined and needs to be filled with a value to make the graph continuous.

Remember, continuity is vital because it allows us to work with functions and analyze their behavior using calculus techniques. So keep practicing, and you'll soon master the concept of continuity!