Equivalence of Definitions of Piecewise Continuously Differentiable Function

Theorem
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Then $f$ satisfies Piecewise Continuously Differentiable Function/Definition 2 iff $f$ satisfies Piecewise Continuously Differentiable Function/Definition 1.

Proof
Assume first that $f$ satisfies the requirements of Definition 2.

We need to prove that $f$ satisfies the requirements of Definition 1.

By Piecewise Continuously Differentiable Function/Definition 2 is Continuous, $f$ is continuous, which proves (1) in Definition 1.

We know from Definition 2 that $f$ is continuously differentiable on $\left[{x_{i−1} \,.\,.\, x_i}\right]$, the derivatives at $x_{i−1}$ and $x_i$ understood as one-sided derivatives, for every $i \in \left\{{1, \ldots, n}\right\}$.

From this follows that $f$ is continuously differentiable at every point of every $\left({x_{i−1} \,.\,.\, x_i}\right)$ since $\left({x_{i−1} \,.\,.\, x_i}\right)$ is a subset of $\left[{x_{i−1} \,.\,.\, x_i}\right]$ where $f$ is continuously differentiable.

This proves the first part of (2) in Definition 1.

Now, $f$ has a continuous right-hand derivative at $x_{i−1}$ for every $i \in \left\{{1, \ldots, n}\right\}$, meaning: $f′_+ \left({x_{i−1}}\right)$ = $\displaystyle \lim_{x \to x_{i−1^+}} f′ \left({x}\right)$, which entails that $\displaystyle \lim_{x \to x_{i−1^+}} f′ \left({x}\right)$ exists for every $i \in \left\{{1, \ldots, n}\right\}$.

Also, $f$ has a continuous left-hand derivative at $x_i$ for every $i \in \left\{{1, \ldots, n}\right\}$, meaning: $f′_- \left({x_i}\right)$ = $\displaystyle \lim_{x \to x_{i^-}} f′\left({x}\right)$, which entails that $\displaystyle \lim_{x \to x_{i^-}} f′\left({x}\right)$ exists for every $i \in \left\{{1, \ldots, n}\right\}$.

These two conclusions prove that the second and last part of requirement (2) in Definition 1 is statisfied.

This finishes the proof that $f$ satisfies the requirements of Definition 1.

Now assume that $f$ satisfies the requirements of Definition 1.

We need to prove that $f$ satisfies the requirements of Definition 2.

We know from Definition 1 that $f$ is continuously differentiable on $\left({x_{i−1} \,.\,.\, x_i}\right)$ and the one-sided limits $\displaystyle \lim_{x \to x_{i−1^+}} f′ \left({x}\right)$ and $\displaystyle \lim_{x \to x_{i^-}} f′\left({x}\right)$ exist for every $i \in \left\{{1, \ldots, n}\right\}$.

Extendability Theorem for Derivatives Continuous on Open Intervals‎ ensures that $f$ is continuously differentiable on $\left[{x_{i−1} \,.\,.\, x_i}\right]$, the derivatives at $x_{i−1}$ and $x_i$ understood as one-sided derivatives, for every $i \in \left\{{1, \ldots, n}\right\}$, and these are exactly the requirements of Definition 2.