Lesson 59 — Differentiability and Smoothness

Let $f(x) = |x|$. Calculate the one-sided derivatives $f'_+(0)$ and $f'_-(0)$ from the definition. Conclude about differentiability at $0$.

Let $f(x) = |x - 3|$. Calculate $f'_+(3)$ and $f'_-(3)$. Is $f$ differentiable at $x = 3$?

Let $f(x) = x|x|$. Determine $f'(0)$ using the definition.

Let $f(x) = x^{1/3}$. Calculate $f'(0)$ from the definition. What does the answer indicate geometrically?

Let $f(x) = x^{2/3}$. Analyze differentiability at $0$ by calculating the one-sided derivatives from the definition.

Let $f(x) = \begin{cases} x^2, & x \geq 0 \\ -x^2, & x < 0 \end{cases}$. Calculate $f'_+(0)$ and $f'_-(0)$. Is $f$ differentiable at $0$?

Let $f(x) = x\sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. Check if $f$ is continuous at $0$ and if it is differentiable at $0$.

Let $f(x) = x^2\sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. Show that $f'(0) = 0$ using the Squeeze Theorem.

Let $f(x) = \max(0, x)$ (ReLU function). Calculate $f'_+(0)$ and $f'_-(0)$. Is $f$ differentiable at $0$?

Let $f(x) = \min(x, 1-x)$. At what point does $f$ have a corner? Verify by calculating the one-sided derivatives at that point.

Let $\text{sgn}(x) = x/|x|$ for $x \neq 0$ and $\text{sgn}(0) = 0$. Is it continuous at $0$? Is it differentiable at $0$?

Where is the floor function $f(x) = \lfloor x \rfloor$ differentiable? Where is it not? Justify in each case.

Let $f(x) = |x|^3$. Calculate $f'(0)$ and $f''(0)$ using the limit definition.

Let $f(x) = x^2$ if $x \in \mathbb{Q}$ and $f(x) = 0$ if $x \notin \mathbb{Q}$. Determine if $f$ is differentiable at $0$.

Let $f(x) = \sqrt{|x|}$. Calculate $f'(0)$ from the definition. Identify the type of non-differentiable point.

Let $f(x) = x^2\sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. Show that $f$ is differentiable at $0$, but that $f' \notin C^0$. What is the maximum $C^k$ class of $f$?

Find $a$ and $b$ such that $f(x) = \begin{cases} ax + b, & x \leq 1 \\ x^2, & x > 1 \end{cases}$ is $C^1$ on $\mathbb{R}$.

Find $c, d$ such that $f(x) = \begin{cases} cx + d, & x \leq 1 \\ 3x^2 - 2, & x > 1 \end{cases}$ is $C^1$ at $1$.

Find $a, b$ such that $f(x) = \begin{cases} ax + b, & x \leq 0 \\ \sin x, & x > 0 \end{cases}$ is $C^1$ at $0$.

Where is $f(x) = (x-2)^{1/3}$ not differentiable? Identify the type of point.

Let $f(x) = \begin{cases} x^2, & x < 0 \\ \sin x, & x \geq 0 \end{cases}$. Is $f$ $C^0$ at $0$? Is it $C^1$ at $0$?

A cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ belongs to which $C^k$ class? Why is the answer not $C^3$?

Where does $f(x) = |x^2 - 1|$ have corners? Calculate the one-sided derivatives at each point to confirm.

A cubic spline is formed by $S_1(x)$ on $[0,1]$ and $S_2(x)$ on $[1,2]$. What conditions at $x = 1$ guarantee the combined spline is $C^2$? List all equations.

Let $f(x) = x|\sin x|$. At which points does $f$ have corners? Sketch the argument for the points $x = n\pi$.

Let $f(x) = x|x|$. Calculate $f'(x)$ for all $x$ and show that $f \in C^1$.

Analyze the differentiability of $f(x) = |\sin x|$ on all of $\mathbb{R}$. At which points does $f$ have corners?

Let $p(x) = 3x^4 - 2x^2 + 7$. What is the $C^k$ regularity class of $p$ on $\mathbb{R}$? Justify.

The payoff of a European call option at expiration is $V(S) = \max(S - K, 0)$. (a) Identify the non-differentiable point. (b) Calculate $V'_-(K)$ and $V'_+(K)$. (c) What happens to the Greek Delta $\Delta = \partial V/\partial S$ at this point?

In machine learning, the ReLU activation function is $f(x) = \max(0, x)$. Why does the SGD algorithm work even though ReLU is not differentiable at $0$?

In structural engineering, a knotted elastic cable has continuous displacement $u(x)$ but a jump in slope $u'(x)$ at the knot. (a) What is the regularity class of $u$? (b) What does the corner in the graph of $u$ represent physically?

A natural cubic spline on $[0,1]$ with a node at $1/2$ imposes what regularity conditions? What is the resulting $C^k$ class? Why $C^2$ and not $C^3$?

For a wave equation $u_{tt} = c^2 u_{xx}$ with a discontinuous initial condition (step function), what regularity is expected for the solution $u(x,t)$? Why does a $C^2$ solution not exist?

Is the converse of "differentiable $\Rightarrow$ continuous" true? What is the simplest counterexample?

Let $f(x) = x|x|$. What is the maximum $C^k$ class of $f$? Calculate $f'(x)$ for all $x$ to justify.

The Cantor function satisfies: continuous on $[0,1]$, $f(0) = 0$, $f(1) = 1$, and $f'(x) = 0$ almost everywhere. Why does the Fundamental Theorem of Calculus not apply?

Does a continuous function exist that is not differentiable at any point? Describe the main construction.

Let $f(x) = e^{-1/x^2}$ for $x \neq 0$ and $f(0) = 0$. Show that $f \in C^\infty$ and that $f^{(n)}(0) = 0$ for all $n \geq 0$. What does this imply about the Taylor series of $f$ at $0$?

Prove: if $f$ is differentiable at $a$, then $f$ is continuous at $a$.

Prove that if $f \in C^1[a,b]$, then $f$ is Lipschitz on $[a,b]$. (Hint: use the Mean Value Theorem and the fact that $f'$ is bounded on a compact set.)