Equations can be classified according to the terms contained in them. Equations that only contain variables raised to a power are polynomial equations. If the highest power is one, then the equation is linear. If the highest power is two, then the equation is quadratic. If the highest power is three then it is cubic. If the equation contains sins, cosines, log or a variety of other functions, then it is called transcendental.
Continuity refers to how well behaved the curve is in a mathematical sense. If, for a value arbitrarily close to a value x0 the function is arbitrarily close to f(x0), then it has positional, or zeroth order continuity at that point. If the slope of the curve (or the first derivative of the function) is continuous, then the function has first order continuity. This is extended to all of the functions derivatives.
If a curve is pieced together from individual curve segments then one can speak of piecewise continuity if each individual segment is continuous and the values at the junctions of the curve segments are continuous.