Can inductor voltage and capacitor current change abruptly?

but can inductor voltage and capacitor current change abruptly?

Yes, in the context of ideal circuit theory. Indeed, this is often the case when there is a switch in the circuit.

Mathematically, if the slope of inductor current (capacitor voltage) changes abruptly, the inductor voltage (capacitor current) is discontinuous.

So, for example, consider the case that a charged capacitor, an open switch, and a resistor are in series (as in problem 2 here)

At the instant the switch is closed, the voltage across the resistor instantaneously changes from zero to the initial capacitor voltage \$V_0\$. Thus, the capacitor current discontinuously changes from zero to non-zero and is given by

$$i_C(t) = \frac{V_0}{R}e^{-\frac{t}{RC}} \cdot u(t)$$

where \$u(t)\$ is the unit step function

The dual of this is an inductor, with non-zero current \$I_0\$, in parallel with a closed switch and a resistor. At the instant the switch is opened, the current through the resistor changes instantly from zero to the initial inductor current. Thus, the inductor voltage discontinuously changes from zero to non-zero and is given by

$$v_L(t) = I_0R\;e^{-\frac{tR}{L}} \cdot u(t) $$

In physical circuits, voltages and currents cannot instantaneously change but depending on the characteristic time scale, they can effectively change instantaneously.

In an ideal world, where a capacitor has no series inductance and an inductor has no parallel capacitance, and voltage and current sources can provide voltages and currents with a step-shaped profile, the current into a capacitor and the voltage over an inductor can change abruptly.

Note that the reverse is not true: the voltage over a capacitor, and the current through an inductor, can not change abrubtly (unless you allow for non-finite currents or voltages, like a Dirac-shaped pulse).

Note that this ideal world is an mathematical abstraction, you can't buy such components.