~Continuity & Differentiability~

$cardinal$

~The following is the basic definition for a function to be continuous at x=a.

~f will be continuous at a if and only if the following limit holds:

lim f(x) = f(a)
x→a

~NOTE: Being CONNECTED is a good way of thinking of the concept of continuity. You will be able to trace the entire curve with your pencil or pen without lifting it off your paper.

~for this limit to hold, 3 conditions must be true:

    i)   f(a) must be defined

   ii)    lim f(x) must exist
          x→a

  iii)    i) = ii)

~these are know as the "3 i's of continuity"

iii) is a necessary & sufficient condition for continuity & some only give
that limit alone, however, to see clearly the types of discontinuity a function might have, one should also look at i) & ii) separately.

~Different types of discontinuities~

1) i) does not hold but ii) does: gives a point (removable) discontinuity

2) i) holds and ii) holds but iii) doesn't : also gives a point discontinuity

   3)   ii) doesn't hold (but the function does not approach +
             or - infinity as x approaches a): gives a jump
             discontinuity (regardless if (i) holds or not).
             In this case, the right and left side limits are not the same.

        ii) doesn't hold ( the function approaches + or - infinity):
            gives an infinity discontinuity (regardless if (i) holds
            or not). In this case, the graph of f has a vertical asymptote.

~The following is the basic definition for a function to be differentiable at x=a.

~ f will be differentiable at a if and only if the following limit holds:

lim [f(x) - f(a)] / (x-a)
x→a

~this is the definition of f ' (a)

~think of being differentiable as being SMOOTH. A point moving on a smooth curve will not encounter any cusps (abrupt changes in slope or vertical tangents)

~for implicitly defined functions, it is possible for the curve to be smooth at a point & have a vertical tangent line there.
(a circle at the end points of a horizontal diameter segment)

~the tangent lines must approach one tangent line at that point having one slope value. If the slopes approach different values from each side of a, a cusp is formed and the function will not be differentiable there.
(not smooth at that point)

~It is possible for the tangent lines to approach one tangent line from both sides and still not have a derivative there. This happens when the cusp contains a vertical tangent line.

~The relationship between continuity & differentiability

~if you have a discontinuity at a point, you cannot be differentiable there.

(if you not connected, you cannot be smooth)

~if you are differentiable at a point, you will be continuous at that point.

(if you are smooth there, then you will be connected there)

~if you are continuous at a point, you may or may not be differentiable there. Connected smoothly gives a derivative, (implicitly defined functions are an exception, but connected with a cusp, does not).