Let f(x) = sin(x) and g(x) = cos(x). By combining f and g, create as many functions as possible that are continuous at every real number x.

We can add or subtract : (f + g)(x) = (g + f)(x) = sin(x) + cos(x) is continuous at every real number x

We can multiply: (fg)(x) = (gf)(x) = (sin x)(cos x) is continuous at every real number x

We can divide: Since sin(x) is sometimes 0 and cos(x) is sometimes 0, dividing either way will result in a function that is not continuous at every real number x. For example, is discontinuous when .

We can compose: (f ο g) = sin(cos(x)) and (g ο f) = cos(sin(x)) are continuous at every real number x.