[Blindmath] Basic mathematical concept question

[Andrew Stacey]

That was a nearly perfect explanation.

Just one small technicality.  The properties that Richard refers to as
"continuous" and "discontinuous" are actually "differentiable" and
"non-differentiable".  The rough-and-ready description of a function being
continuous is that its graph can be drawn without taking the pencil off the
paper.  The actual definition is a little more complicated, but that's the
basic idea.  A continuous function has no jumps, no gaps.  It is
a differentiable function that has a well-defined slope everywhere.

A differentiable function is automatically continuous, but not vice versa.

Some examples:

1. A discontinuous function: define f(t) = 0 if t < 0 and f(t) = 1 otherwise.
This has a jump at 0 so is discontinuous.

2. A continuous function that isn't differentiable: f(t) = 0 if t < 0 and f(t)
= t otherwise.  This is continuous, but it has no well-defined slope at
0 since as we approach 0 from the left the slope appears to be 0 (flat)
whereas if we approach it from the right then the slope appears to be 1.

Most functions that you encounter are differentiable, in fact probably even
better than that.  But properly defined, most functions that exist are not
differentiable anywhere.  It's just that since differentiable functions are so
easy to study (compared to arbitrary functions), we use them if at all
possible.

Andrew Stacey