# Continuous Function at a Point? Based on Limits

• Mar 30th 2011, 07:46 AM
Savior_Self
Continuous Function at a Point? Based on Limits
http://i55.tinypic.com/15dmf6a.jpg

The Definition of Continuity at a Point is "A function of f is continuous at a point c is f exists at c and (the limit of f(x) as x approaches c from the left) = (the limit of f(x) as x approaches c from the right) = (f(c))"

My problem with this is I don't know enough to know whether they're trying to trick me by omitting the knowledge if the two limits equal f(1) and, because that's omitted, I should answer with "The point at f(c) does not necessarily exist." or what..

Could someone explain this to me? Thanks.
• Mar 30th 2011, 07:54 AM
TheEmptySet
Quote:

Originally Posted by Savior_Self
http://i55.tinypic.com/15dmf6a.jpg

The Definition of Continuity at a Point is "A function of f is continuous at a point c is f exists at c and (the limit of f(x) as x approaches c from the left) = (the limit of f(x) as x approaches c from the right) = (f(c))"

My problem with this is I don't know enough to know whether they're trying to trick me by omitting the knowledge if the two limits equal f(1) and, because that's omitted, I should answer with "The point at f(c) does not necessarily exist." or what..

Could someone explain this to me? Thanks.

You are correct the limits tell you nothing about the values at the point. For example

$f(x)=\begin{cases}x^2, \text{ if } x \ne 0 \\ 29, \text{ if } x=0 \end{cases}$

Both the right and left limit equal zero, but $f(0)=29$
• Mar 30th 2011, 08:19 AM
HallsofIvy
In other words, the statement is false.
• Mar 30th 2011, 08:40 AM
bkarpuz
A simple counter example
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