Cases in which the limiting value of a function $f(x)$ (as $x\to c$) is not equal to $f(c)$? Can anyone state some cases such as mentioned in the title. I have tried to look for them on google and my textbook, but cant find any examples. And incidentally how can this fact be ever true? does it have something to with the function not  being defined at the limiting point $c$?

There are artificial examples created by piecewise definitions. (Richard Feynman once expressed surprise that anybody thought such things are functions.)  I'll try to give a more serious example: You are walking past a building with a conventional rectangular shape.  It is on your left.  The distance from you to the rightmost point on the building that you can see changes in a continuous manner until you pass the corner, and then it abruptly increases; it has a jump discontinuity.
