Consider a beam that is arbitrarily loaded and at any distance x, we take an element in the beam between two sections, where a point force acts as shown.
Let us assume the directions of shear force and bending moment to be as shown in the figure.
Applying the equation of equilibrium we get,
$$V_1-P-V_2=0$$ $$V_2=V_1-P$$
Thus, $V_x=+V_1$ and $V_{x+dx}=+(V_1-P)$
Note: Shear force which tries to rotate the element clockwise is taken as +ve and Bending moment which tends to compress the upper part of the beam is taken as positive.
We conclude that the shear force abruptly changes as we go from left to right across the point force.
Taking moment about a point on the right face of the element:
$$-M_1-V_1dx+Pdx/2+M_2=0$$
$$M_2=M_1+V_1dx-Pdx/2$$
Thus, $M_x=+M_1$ and $M_{x+dx}= +(M_1+V_1dx-Pdx/2)$
The change in the bending moment $M_2-M_1$ is differential, thus as we go from left to right across a point force the bending moment essentially remains constant.
My trouble starts from here:
If I try determining the slope of the BMD at x,
$$\frac{dM}{dx}=\frac{M_{x+dx}-M_x}{dx}=\frac{+(M_1+V_1dx-Pdx/2)-+M_1}{dx}=V_1-P/2$$
However the book I'm referring to says that - "Even though the bending moment M does not change at a concentrated load, its rate of change dM/dx undergoes an abrupt change".
I don't get how the slope undergoes an abrupt change, as from the analysis the slope is turning out to be $$V_1-P/2$$ at x.