Continuity:

A function f is continuos at x = a if f is defined at x = a and either ; f is not defined anywhere near a; or f is defined arbitrarily near x = a and lim_x®a f(x)=f(a).

Conversely, A function f is discontinuous at x = a if f is defined at x = a and f is not continuos at x = a.

Addition, Subtraction, Multiplication and Division of Continuity:

If f and g are continuos functions at x = a then; f + g, f - g, f · g and f/g; where g(a) ¹ 0; are also continuous at x = a.

If the function f is a polynomial or a rational function then f is continuos wherever it is defined.

Given f(g(x)), where g is continuos at x = a , and f is continuos at x = g(a) then f(g(x)) is continuos at x = a.