Work-Energy Theorem for a path that is not smooth

Question

In the analysis of Newtonian Mechanics for a single particle, we come across the definition of work and also the Work-Kinetic Energy theorem:

For a single particle, the work done on a particle by a force $\textbf{F}$ in transforming the particle from condition A to condition B is defined as: $$W_{AB} = \int_A^B{\textbf{F}}.d\textbf{r}$$ and the integrand can be reduced to an exact differential, if $\bf{F}$ is the net force acting on the particle, as: $${\textbf{F}}.d\textbf{r} = m.\frac{d\textbf{v}}{dt}\textbf{.}\frac{d\textbf{r}}{dt}.dt = m\textbf{v}.d\textbf{v} = dK \implies \boxed{W_{AB} = K_B - K_A}$$ However, can the same theorem hold if the curve $\textbf{r} = \textbf{r}(t)$ is such that there are points where $\textbf{r}(t)$ is not differentiable, such as path 1 in the picture below?

(1) In such a case, $\frac{d\textbf{r}}{dt}$ would not be defined at the sharp point, so would the theorem still hold?

(2) How would you evaluate work done along path 1 given that $$\int_1 \textbf{F}.\textbf{e}_{t} \ ds$$ is not defined at the same point? [Here, $\textbf{e}_{t}$ is the unit tangent vector, and $ds$ the arc length parameter, such that $d\textbf{r} =\textbf{e}_{t}.ds]$

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Answer 1

However, can the same theorem hold if the curve $\textbf{r} = \textbf{r}(t)$ is such that there are points where $\textbf{r}(t)$ is not differentiable, such as path 1 in the picture below?

Yes.

(1) In such a case, $\frac{d\textbf{r}}{dt}$ would not be defined at the sharp point, so would the theorem still hold?

Yes.

(2) How would you evaluate work done along path 1 ...

Split it into two straight line contributions: $$ \int_{s=0}^{s=1} F_y(x_0, y_0 + \Delta_1 s)ds + \int_{s=0}^{s=1} F_x(x_1 + \Delta_2 s, y_1)ds\;. $$

Answer 2

You do the same as at most discontinuities which feature in ideal situations.

Evaluate $\displaystyle \int_1 \textbf{F}.\textbf{e}_{t} \ ds$ in two parts, $ \displaystyle \int_A^B \textbf{F}.\textbf{e}_{t} \ ds $ and $\displaystyle \int_D^E \textbf{F}.\textbf{e}_{t} \ ds$ with the discontinuity at position $C$ and making positions $B$ and $D$ as close as possible (tending to) position $C$.