A graph is said to be continuous if it is entirely smooth and curved. It will have no sudden vertical jumps (upwards or downwards) or holes as you follow the line along the x-axis (horizontal axis), and no sudden changes in direction that make it jagged rather than curved.
A graph that is discontinuous however has at least one such of these vertical jumps, holes, or sudden changes in direction somewhere along it. The graph is discontinuous at the places on the graph where the jumps/holes/direction changes are and continuous elsewhere. The continuous sections of the graph are called segments or pieces. A discontinuous graph, although not fully continuous, can be piecewise continuous if it has segments/pieces that are continuous over certain intervals on the x-axis.
The mathematical property that discontinuities on a graph may have is that the value of the graph (on the y-axis) at the discontinuity is different depending on whether you approach it from the left or from the right. The gradient at a discontinuity is undefined so that the graph is not differentiable at that point, but if the graph is left- or right-continuous at the continuity the gradient may tend to zero or infinity from the direction it is continuous in, but jump suddenly from the opposite direction. A continuous graph on the other hand is infinitely differentiable, in that you can keep on differentiating again and again and again and always get a number.
For example, straight line graphs of the form y = ax + b are continuous as there are no jumps, holes, and the gradient is constant (and equal to a). The graph of y = tan(x) however is discontinuous, having a discontinuity at the end of its phase where from the left the gradient tends to infinity, and from the right it tends to infinity - the graph looks like it shoots upwards off the page and returns directly from the bottom of the page, to infinity, beyond, and then back to minus infinity. The graph y = tan(x) is piecewise continuous in each of its repeated phases, which periodically repeat forever in both directions on the x-axis.
Bar graphs are also discontinuous in nature as there is a vertical jump at each new bar. The gradient is constant within a bar (and equal to zero), and the graph is piecewise continuous over each of the bars on the graph.
Hybrid graphs that take on different forms depending on the zone of the x-axis they inhabit can be piecewise continuous if the segments in each zone or interval on the x-axis are continuous in themselves. If you had a line graph up to a certain point on the x-axis and then another line graph beyond that point with a different slope this would be piecewise continuous as the two pieces (the two straight lines) are continuous as individual pieces.
Line graphs y = ax + b are an example of a continuous function. The trigonometric function y = tan(x) is an example of a discontinuous function as it has a discontinuity at the end of its periodic phase.
Continuous Function: If you can draw a function without lifting the pencil from your paper, its continuous! So you can imagine that it would be a smooth, examples include parabolas (F(x))=x^2)
*However: there are more complex curves where you need to check for 3 important conditions~
1) Does the lim x-->c exist?
2) Does the F(c) exist?
3) Does the lim x-->c equal f(x)?
Differentiable Function: When you think of differentiable think derivative. In this function the derivative exists at all values in its domain. Similarly to a continuous graph it will be a smooth function as well. The lim as x approaches c from BOTH the positive and negative side will exist.
*Keep in mind that if a function is continuous, it can be differentiable.
*If a function is NOT continuous, it cannot be differentiable.
Main Source: AP Calculus REA AB/BC