What is a continuous function?

1 Answer
Sep 13, 2015

A continuous function is a function that is continuous at every point in its domain.

That is #f:A->B# is continuous if #AA a in A, lim_(x->a) f(x) = f(a)#

Explanation:

We normally describe a continuous function as one whose graph can be drawn without any jumps. That's a good place to start, but is misleading.

An example of a well behaved continuous function would be #f(x) = x^3-x#

graph{x^3-x [-2.5, 2.5, -1.25, 1.25]}

In fact any polynomial is well defined everywhere and continuous.

A less obvious example of a continuous function is #f(x) = tan(x)#

graph{tan(x) [-10, 10, -5, 5]}

This appears to be discontinuous, with 'jumps' at #x = pi/2+n pi# but those values of #x# are excluded from the domain.

Similarly, the following function is continuous on its domain #(-oo, 0) uu (0, oo)#

#f(x) = x/abs(x)#

graph{x/abs(x) [-5, 5, -2.5, 2.5]}

If we add a definition of #f(0)# then this becomes a discontinuous function.

#f(x) = { (x/abs(x), "if x != 0"), (0, "if x = 0") :}#

graph{(y-x/abs(x))(x^2+y^2-0.002) = 0 [-5, 5, -2.5, 2.5]}

This #f(x)# fails the condition for continuity at the point #x=0#.

#lim_(x->0) f(x)# is not defined, let alone equal to #f(0)#