What is a continuous function?

1 Answer
Jul 9, 2015

Answer:

There are several definitions of continuous function, so I give you several...

Explanation:

Very roughly speaking, a continuous function is one whose graph can be drawn without lifting your pen from the paper. It has no discontinuities (jumps).

Much more formally:

If #A sube RR# then #f(x):A->RR# is continuous iff

#AA x in A, delta in RR, delta > 0, EE epsilon in RR, epsilon > 0 :#

#AA x_1 in (x - epsilon, x + epsilon) nn A, f(x_1) in (f(x) - delta, f(x) + delta)#

That's rather a mouthful, but basically means that #f(x)# does not suddenly jump in value.

Here's another definition:

If #A# and #B# are any sets with a definition of open subsets, then #f:A->B# is continuous iff the pre-image of any open subset of #B# is an open subset of #A#.

That is if #B_1 sube B# is an open subset of #B# and #A_1 = { a in A : f(a) in B_1 }#, then #A_1# is an open subset of #A#.