a mathematical function is called continuous if, roughly said, a "small change" in the input only causes a "small change" in the output. if this is not the case, the function is discontinuous. functions defined on the real numbers, with one input and one output variable, will show as an "uninterrupted line (or curve)". they can be drawn "without lifting the pen off of the page". the definition given above was made by augustin-louis cauchy.