Continuous Function:

A function is said to be continuous if there is no abrupt changes or discontinuity in the function. The best way to measure the continuity of a function at any point is to approach from both sides of the function to that point. If you reach at the same point or if you get the same value of the function, then the function is continuous at that point.

For example, in the following figure, the parabola is continuous at every point.

Let us take a point x= 0 to test the continuity. If we approach from right side toward  x= 0 and from left side toward x= 0, we get the same value of the function which is 0. Similarly we can prove that for all points, this is a continuous function.

Piecewise Continuous Function:

Piecewise continuous function is a function which is a piecewise function and continuity exists at every point. For example, the following function f(x) cannot be described with one equation rather it is a combination of three equations for different x value. So it is a piecewise function and since continuity exists at every point of the function, it is piecewise continuous function. Following figure shows a piecewise continuous function described by,

y = \begin{cases} x&0\leq x \leq 3 \\ 3&3\leq x \leq 6 \\ -x + 9&6\leq x\leq9 \end{cases}

Discontinuous Function:

A function is said to be discontinuous if there is abrupt changes or discontinuity in the function. In the following figure, the function is discontinuous at x=3 because if we approach from left and right toward x= 3, we reach at two different points or we get two different values of the function. For   0\leq x \leq 3 \quad  y = x and for x>3 \quad y= 7.  So this is a discontinuous function.

Discrete Function:

A discrete function x(n) is a function of an independent variable, n which is integer. A discrete function is not defined between successive integers. Also it is incorrect to say that x(n) is equal to zero if n is not integer. For example a discrete function x(n) is defined as x[n] = \{3,2,\uparrow{1},2,3\} . Here uparrow with 1 indicates at zero value of function is 1. This function can be described by following diagram.