Continuity of Function?

They are discontinuous at the "steps" but continuous everywhere else. The derivatives would exist on the continuous sections.

I'm not familiar with the term step-wise; perhaps it is one of these:

Step functions, those functions which attain a countable number of discrete values (on intervals) are discontinuous almost by definition. An example:

f(x) = 0, x > 0, = 1, x <= 0.

In these functions the dertivatives exist everywhere but the junctions.

Piecewise [defined] functions are those (including step functions) which are defined by means of several formulae. These can be continuous (and even differentiable) at the juctions. An example:

f(x) = x^2, |x| > 0, = 0, x = 0.

These sorts of functions will be continuous whenever the value at the junction is equal to the limit of the funciton at the junction (differentiability requires a bit more).

Hope this helps.