Question

What are the the three things a continuous function can't have?

Answer 1

There are more than three, but I'll try: In order for a function to be continuous on `(-oo,oo)` it can have no holes, jumps and vertical asymptotes.

Explanation:

I am taking as a definiton:

`f` is a continuous function if and only if , for every real number, `a`, `f` is continuous at `a`

and `f` is continuous at `a` if and only if `lim_(xrarra)f(x) = f(a)` (existence implied).

In order for `f` to be continuous at `a`,

`f(a)` must exist, so there can be no hole at `x=a`

`lim_(xrarra) f(x)` must exist which means no jump (different one-sided limits) and no vertical asymptote at `x=a`.

Those are the first 3 I thought of, but the list in not complete. A continuous function also cannot have the kind of "infinite discontinuity" that

`f(x) = {(sin(1/x),if,x != 0),(0,if,x = 0) :}`

has at `0`. The limit does not exist, but there is no jump and no vertical asymptote.

Here is most of the graph of `f(x) = {(sin(1/x),if,x != 0),(0,if,x = 0) :}`.

(I can't get Socratic's graphing utility to put `(0,0)` on the graph.)

graph{(y-sin(1/x))=0 [-0.743, 0.756, -1.2, 1.2]}