I'm a fan of hidden case-wise formulas. Principle building block in this world is the absolute value function $|x| = \sqrt{x^2}$. Already, we have a function which is continuous, but, not differentiable at $0$. This gives a counter-example for a misuse of the Mean Value Theorem: if $f(x) = \sqrt{x^2}$ then $f(-1)=f(1)=1$ yet, nowhere in $[-1,1]$ do we find zero derivative. Oops, it seems I just gave an example of Rolle's Theorem failing. Not really. Of course $f$ is not differentiable everywhere in $(-1,1)$. The formula seems innocent enough, it's just the square and root function. Differentiation yields another misbehaving function: $$ \frac{d}{dx}\sqrt{x^2} = \frac{2x}{2\sqrt{x^2}} = \frac{x}{\sqrt{x^2}}$$ Let $g(x) = \frac{x}{\sqrt{x^2}}$ then this function is discontinuous at $x=0$ but elsewhere constant. Note, $g'(x)=0$ for $x \neq 0$, yet, $g$ is not globally constant. This weirdness is possible since the domain of $g$ is disconnected.

In Penrose's Road to Reality he studies the function $h(x) = x|x| = x \sqrt{x^2}$. This gives us a differentible function which is not twice differentiable at $x=0$. Indeed, the function $f_n(x) = x^n|x|$ is an easy example of the function which has $n$-derivatives at $x=0$, but, the $n$-th derivative is discontinuous at $x=0$.

I'm not quite sure what you're after, but, I think I've made the case for the absolute value function. We have a simple formula, but, simple twists on it give rather varied behaviour.

Beyond this, I also think it is interesting to study the interplay between vertical and horizontal behaviour of the function and its inverse function. For example, horizontal tangents of sine translate to vertical tangents of inverse sine. Or, just the simple idea of graphing the same shape horizontally verses vertically. The connection between the slopes, increasing paired with increasing, or decreasing with decreasing. Much to explore here there is.

I'm a fan of hidden case-wise formulas. Principle building block in this world is the absolute value function $|x| = \sqrt{x^2}$. Already, we have a function which is continuous, but, not differentiable at $0$. This gives a counter-example for a misuse of the Mean Value Theorem: if $f(x) = \sqrt{x^2}$ then $f(-1)=f(1)=1$ yet, nowhere in $[-1,1]$ do we find zero derivative. Oops, it seems I just gave an example of Rolle's Theorem failing. Not really. Of course $f$ is not differentiable everywhere in $(-1,1)$. The formula seems innocent enough, it's just the square and root function. Differentiation yields another misbehaving function: $$ \frac{d}{dx}\sqrt{x^2} = \frac{2x}{2\sqrt{x^2}} = \frac{x}{\sqrt{x^2}}$$ Let $g(x) = \frac{x}{\sqrt{x^2}}$ then this function is discontinuous at $x=0$ but elsewhere constant. Note, $g'(x)=0$ for $x \neq 0$, yet, $g$ is not globally constant. This weirdness is possible since the domain of $g$ is disconnected.

In Penrose's Road to Reality he studies the function $h(x) = x|x| = x \sqrt{x^2}$. This gives us a differentible function which is not twice differentiable at $x=0$. Indeed, the function $f_n(x) = x^n|x|$ is an easy example of the function which has $n$-derivatives at $x=0$, but, the $(n+1)$-th derivative does not exist at $x=0$.

I'm not quite sure what you're after, but, I think I've made the case for the absolute value function. We have a simple formula, but, simple twists on it give rather varied behaviour.

Beyond this, I also think it is interesting to study the interplay between vertical and horizontal behaviour of the function and its inverse function. For example, horizontal tangents of sine translate to vertical tangents of inverse sine. Or, just the simple idea of graphing the same shape horizontally verses vertically. The connection between the slopes, increasing paired with increasing, or decreasing with decreasing. Much to explore here there is.