Definition: A function f is continuous at a number a if lim _x->a f(x) = f(a)

If f is not continuous at a, then it is discontinuous at x=a. Note we have three requirements

• f(a) must be defined, i.e. a must be in domain of f
• lim _x->a f(x) must exist
• lim _x->a f(x) must equal f(a)

Definition: A function f is continuous from the right at a number a if lim _x->a⁺ f(x) = f(a) and continous from the left at a number a if lim _x->a^- f(x) = f(a) A function is continuous on an interval (a,b) if it is continuous for every number in the interval.

Theorem If f and g are continuous at x=a, and if c is a constant then the following are also continuous at x=a:

1. f+g
2. f-g
3. cf
4. f.g
5. f/g , if g(a) does not vanish

Theorem

a polynomial is continous everywhere

a rational function is continous every it is defined

Theorem If f is continuous, lim _x->a f(g(x)) = f (lim _x->a g(x))

Theorem If f and g are continuous, then (fog)(x) is also.

Intermediate Value Theorem If f is continous on [a,b] and N lies between f(a) and f(b), then for some number c in (a,b), f(c) = N

A vector valued function r(t)= is continuous at t=a if its components are.