A function f(x) is continuous at a point a if it behaves as it should there, that is:
Definition 1 Assume that f(x) is defined in some open interval that contains a . Then, f is continuous at a if: lim x® a f(x) = f(a). f is continuous on the interval (b,c) if it is continuous at all points a Î (b,c) .
Note that this says two things, that the limit exists, and that the function's value at a is the same as the limit.
The usual idea of continuity is that a graph is continuous if you can draw the picture without lifting your pencil from the paper. That's a little limited, but gives the idea well enough.
A similar definition can be made for continuity from the left.
Example 1 Let f(x) = ìí î 3x+1, if x ³ 1 x+2, if x < 1 .
Example 2 Let f(x) = ìí î x2, if x ¹ 1 5, if x = 1 . This function is also not continuous at x = 1 , since lim x® 1 f(x) = 1, even though f(1) = 5 . The difference here is that the function could have been continuous, if it weren't for the fact that it was ``poorly'' defined at x = 1 .
Example 3 Let f(x) = ìí î 3x+1, if x ³ 1 x+3, if x < 1 . Then f is continuous at a = 1 . This is because lim x® 1 f(x) = 4 does exist, and is the same as f(1) .
Example 5 f(x) = |x| is continuous, even though it has ``bad behavior'' at x = 0 . Its behavior is just not that bad there. The limit of f(x) , as x approaches 0, is 0, and since that is the value of the function, then it is continuous there. At other points, f is just like a polynomial in some open interval (either x or -x ), so it is continuous.
Exercise 1 Find the values of a for which the function f(x): = ìí î x2-2x, if x £ 2 ax+4, if x > 2, will be continuous at x = 2 . Answer: a = .
Exercise 2 Is there a way to define the value of f(0) so that the function f(x) = x|x| will be continuous at x = 0 ? Answer: f(0) = .
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Copyright (c) 2000 by David L. Johnson. File translated from TEX by TTH, version 2.61.On 18 Oct 2000, 00:06.
Copyright (c) 2000 by David L. Johnson.