what was the function of the Roman cavalry?

[Aryaman2]

Col. Ardant du Picq argues that cavalry were ineffective against steady and disciplined ranks of infantry, and he was an actual contemporary of, and witness to, cavalry in battle (9th Chausseurs and the Crimean).

In other periods cavalry did charge steady infantry, often overruning it, and it was trained to do it.

In what periods for instance?

For instance in the 17th century, John Cruso in his "Militarie Instructions for the Cavallerie", the standard manual used for cavalry in the English Civil War, wrote how to train a horse "to ride him against a compleat armour, so set upon a steak, that he may overthrow it, and trample it under his feet: that so (and by such other means) your horse (finding that he receiveth no harm) may become bold to approach any object"
In that period there are plenty of battles in which infantry was charged and routed by cavalry.