When is $f''(x)$ undefined but $x$ not an inflection point of $f$?

Question

This is supposed to be for bullet 5.2 in this question: A rigorous yet intuitive summary of inflection and critical points for beginning calculus?

• There are examples for undulation points, where $f: A \to B, A,B \subseteq \mathbb R, f''(x)=0$ but $(x,f(x))$ is not an inflection point of $f$ like $f(t) = \frac{t^4}{12},f''(t)=t^2$ (If $f''(x)=0$ but is not an inflection point, what is it called?)

• There are examples where $f''(x)$ is undefined but $(x,f(x))$ is an inflection point like $f'(t)=|t|$ (An inflection point where the second derivative doesn't exist?)

I'm missing something very obvious, but what's an example where $f''(x)$ is undefined, but $(x,f(x))$ is not an inflection point of $f$?

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After some thought, I have come up with some examples. Are these correct?

First example: This example is one where $f$ is defined but not continuous at $x$.

• $f: \mathbb R \to \mathbb R, f(t)=\text{sign}(t), x=0$. $f'(0)$ is undefined and thus so is $f''(0)$. However $f''(t)=0$ for $t \ne 0$.

• In case someone asks further 'Okay, what if we assumed $f$ is continuous? Then $x$ would have to be an inflection point right?', then we proceed:

Second example: This example is one where $f$ is continuous but not differentiable at $x$.

• A sharp turn like $g(t)=|t|$ but always increasing at an increasing rate but at different rates $f: \mathbb R \to \mathbb R, f(t)= t 1_{t \le 0} + 3t 1_{t \ge 0}$. This gives us $f': \mathbb R \setminus \{0\} \to \mathbb R, f'(t) = \text{sign}(t)+2$. $f'(0)$ is undefined and thus so is $f''(0)$. However $f''(t)>0$ for $t \ne 0$.

• In case someone asks further 'Okay, what if we assumed $f$ is differentiable? Then $x$ would have to be an inflection point right?', then we proceed:

Third example: This example is one where $f$ is differentiable at $x$.

• $f': \mathbb R \to \mathbb R, f'(t)= t 1_{t \le 0} + 3t 1_{t \ge 0}$.

Answer 1

I can’t tell what you're asking, so I’ll answer the question in the title. Or, more precisely, I’ll give an example of an $f$, differentiable on the whole real line, with the property you describe in the title. I can’t tell what you’re doing by listing all those cases.

Take $f$ to be $x^2$ when $x$ is negative and $x^4$ when $x$ is non-negative.

Then $f’’(0)$ is undefined, and moreover $f$ does not have an inflection point at the origin.