Recall that a function  is piecewise continuous on [a, b] if

1.  and  are finite.
2.  is continuous at all but possibly finitely many points of [a, b].
3. At any point in (a, b) where  is discontinuous, the function has finite left and right limits.

The figure below illustrates the above points.

At points in (a, b) where the function is discontinuous, the graph has jump discontinuities. The magnitude of the jump at  is the difference between the left and right limits there:

This is the distance between left and right “ends” of the graph at . We will denote left and right limits as follows:

 and

If  has a right limit at , the right derivation of  at  is defined to be

when this limit exists and is finite. We interpret  as the slope of the tangent to the graph at  if we look at only the portion of the graph to the right of . The left derivative of  at  is

when this limit exists and is finite. This number is the slop of the graph of  at , looking at only that portion to the left of . This is illustrated below.

We see that the left and right derivations of a function are given by the gradient of the respective dotted lines. At a jump discontinuity, the left and right derivatives may exist, while the derivative does not exist. Think of it as the graph having a tangent “from the left” and “from the right” at  while having no tangent line at .

Some notes of evaluating these limits. Currently, we seem to be taking the limiting values when we approach integers from left or the right. Notice carefully that when evaluating the left and right limits, we let h tend towards 0 from the positive side alone but when finding the left and right derivatives, we let h tend towards 0 from the positive side for the left derivative and from the negative side for the right derivatives. While it may seem trivial now, after all h tends towards 0 anyways, I believe we should maintain the mathematical rigor throughout the lessons, even if the importance of which may seem irrelevant at present.