A function “f” in interval [a,b] is said to be a continuous function when the Graph drawn for f(x) is a smooth line or curve without any break in it.

Such curve or line can be drawn by the continuous motion of a pencil in a sheet of paper.

And Discontinuous  function is just opposite of the continuous function , the function “f” is said to be  discontinuous function when the graph drawn for f(x) is  consists of disconnected curves or lines.

For example:

Continuous Function:

Discontinuous Function:

If we zoom into the disconnected place of two curves it looks like:

When x_a is any point in the  interval [a,b] then the following relation should be true in the curve for the function “f” to be a continuous function if the following relation is not true in the curve then it is a discontinuous function.