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

1 Answer
Nov 23, 2016

Answer:

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]}