What does it mean for a function to be differentiable? Instead of using
mathematics, how about simply saying, a function is differentiable at a point
if, as you zoom in on that point on the curve, the curve around it looks more
and more like a straight line.
For example, consider the function
Figure 1 shows the plot of this function on the range [-1.5, 6.5], while Figure 2
shows a zoom of Figure 1 on the point point (2.5, f(2.5)). Once you zoom in by a
factor of 28528, the wild curve appears to be quite smooth.
If you were to calculate the derivative
of f(x), you would note that f(1)(2.5) ≈ 1.087766180 and
looking at Figure 2, you will note that the slope, after zooming, is slightly
greater than 1.
Figure 1. The function f(x) plotted on the interval [-1.5, 6.5].
Figure 2. A zoom on the function f(x) at the point (2.5, f(2.5)).
Therefore, if zooming in sufficiently closely produces something which
looks like a straight line, it makes sense that we could use a formula like
to estimate the slope at x and as h → 0, the approximation
should approach a constant value.
Now, you can probably guess that zooming in on the absolute value function |x|
at 0 will not result in a straight line: it will always have a right-angled bend
at 0. Note, however, that there are well-defined functions
which are continuous everywhere and differentiable nowhere. For an example of such
a beast, click here. Another monster is a
function which is continuous at all of the irrationals, but discontinuous at all
of the rationals.