What's the difference between continuous and piecewise continuous functions? A continuous function is a function where the limit exists everywhere, and the function at those points is defined to be the same as the limit. 
I was looking at the image of a piecewise continuous function on the following page: http://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx 
But the image of the function they've presented isn't continuous. As such, I'm confused by what a piecewise continuous function is and the difference between it and a normal continuous function.
I'd appreciate it if someone could explain the difference between a continuous function and a piecewise continuous function. Also, please reference the image of the piecewise continuous function presented on this page http://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx .
Thank you.

A piece-wise continuous function is a bounded function that is allowed to only contain jump discontinuities and fixable discontinuities. These functions almost always occur with the inclusion of floor into the regular set of algebraic functions you are used to in calculus. The reason for this is because up until then there are no functions you have encountered containing any form of jump discontinuity of the finite nature. As the other answer here says, each interval is continuous. This is true. However there are levels of piece-wise continuity which simply put mean that the function is differentiatable fully on those continuous intervals n number of times.
