Start from the name. Caspases are cysteine proteases that cut their targets specifically after aspartate residues. That single property is the engine of programmed cell death.
In apoptosis a cascade runs in sequence. Upstream initiator caspases switch on in response to death signals, and they in turn activate downstream executioner caspases. These executioners chop up structural proteins, the nuclear lamina, and DNA-repair enzymes, neatly packaging the dying cell into membrane-bound apoptotic bodies that are swept up by phagocytes without leaking contents.
The other answers do not fit. Cell division runs on the cyclin and CDK machinery. Necrosis is messy, passive death from injury and swelling, with no controlled protease cascade. Inflammation is the work of cytokines and immune cells; one caspase touches that world by processing a cytokine, but the textbook association demanded here is cell suicide.
So the function that defines caspases is apoptosis.
\[\boxed{\text{Apoptosis (option C)}}\]