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.