Differentiability of a function indicates its smoothness

A function that is differentiable is said to be smooth of degree $1$ , and function that has $k^{th}$ order derivatives is said to be smooth of degree $k$ . Intuitively, everyone has a notion of smoothness of a curve. Does differentiability somehow characterize the same intuitive notion? If so, how? We will try to answer these questions here.

Intuitively, a curve is a smooth if it does not have sharp points. For example:

A straight line is as smooth as any curve can get
The function $f(x) = e^x$ is smooth
The following function is not smooth at $x = 0$ :
$f(x) = \begin{cases} x + 1 & \mbox{ if }x \le 1 \\ -x + 1 & \mbox{ if }x > 1\end{cases}$

We say a curve is smooth at a point if it is not sharp – that is, it resembles a straight line, at least locally. Now, how does $f(x) = e^x$ resemble a straight line at any point? For example, take the point $(1, e)$ on the curve. If we gradually zoom into the curve at this point, we will see its curvature disappear and it appearing more and more like a straight line. How about the function in example $3$ above? No matter how much we zoom in at $x = 0$ , the function will show the sharp point – it will not show a straight line.

The above intuitive notion is what is exactly captured by the differentiability of a function. Derivative of a function $f(x)$ at $x = x_0$ is defined as $\lim \limits_{h \to 0}\frac{f(x_0+h) - f(x_0)}{h}$ .

Now, $\frac{f(x_0+h) - f(x_0)}{h}$ is the slope of a line going through the points $(x_0, f(x_0))$ and $(x_0+h, f(x_0+h))$ . If we say a function is differentiable, $\frac{f(x_0+h) - f(x_0)}{h}$ approaches a limit value as $h$ approaches zero.

Take the sequence of $\{h_n = \frac{(-1)^n}{n}\}$ that approaches zero. This sequence in turn gives a sequence of points $\{(x_0+h_n, f(x_0+h_n)\}$ and sequence of lines for which one point is fixed at $(x_0, f(x_0))$ and another point is $(x_0+h_n, f(x_0+h_n)$ . Note that because of our choice of $h_n$ ‘s, the sequence of lines is such that while each of the lines is “anchored” at $(x_0, f(x_0))$ , the lines alternate for the other point between the left side and the right side of $(x_0, f(x_0))$ . Still, the slopes of the entire sequence approaches a limit value. Since one point is fixed on the line and the slopes approach a limit value, the sequence of lines themselves approach a straight line. That is, for each point in a neighborhood of $(x_0, f(x_0))$ (meaning, points close enough to $(x_0, f(x_0))$ , but on either side of it), each of the lines going through that point and $(x_0, f(x_0))$ have small differences in slope. Thus, the curve appears like a straight line locally.

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