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Chap 4 Continuity

Chap 4 Continuity - FIRST YEAR CALCULUS W W L CHEN c W W L...

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FIRST YEAR CALCULUS W W L CHEN c W W L Chen, 1982, 2005. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 4 CONTINUITY 4.1. Introduction Example 4.1.1. Consider the function f ( x ) = x 2 . The graph below represents this function. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. Example 4.1.2. Consider the function f ( x ) = x/ | x | , as discussed in Example 3.1.6. If we now attempt to draw the graph representing this function, then it is impossible to draw this graph without lifting our pencil from the paper. After all, there is a break, or discontinuity, at x = 0, where the function is not defined. Even if we were to give some value to the function at x = 0, then it would still be impossible to draw this graph without lifting our pencil from the paper. It is impossible to avoid the jump from the value 1 to the value 1 when we go past x = 0 from left to right. Chapter 4 : Continuity page 1 of 9
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First Year Calculus c W W L Chen, 1982, 2005 Example 4.1.3. Consider the function f ( x ) = x 3 + x . We showed in Example 3.1.9 that f ( x ) f (1) as x 1. The graph represents this function. As we approach x = 1 from either side, the curve goes without break towards f (1). In this instance, we say that f ( x ) is continuous at x = 1. We observe from Example 4.1.3 that it is possible to formulate continuity of a function f ( x ) at a point x = a in terms of f ( a ) and the limit of f ( x ) at x = a as follows. Definition. We say that a function f ( x ) is continuous at x = a if f ( x ) f ( a ) as x a ; in other words, if lim x a f ( x ) = f ( a ) . Example 4.1.4. The function f ( x ) = x 2 is continuous at x = a for every a R . Example 4.1.5. The function f ( x ) = x/ | x | is continuous at x = a for every non-zero a R . To see this, note that for every non-zero a R , there is an open interval a 1 < x < a 2 which contains x = a but not x = 0. The function is clearly constant in this open interval. Example 4.1.6. The function f ( x ) = x 3 + x is continuous at x = a for every a R . Example 4.1.7. The function f ( x ) = sin x is continuous at x = a for every a R . To see this, note first the inequalities | sin x sin a | = | 2cos 1 2 ( x + a )sin 1 2 ( x a ) | ≤ | 2sin 1 2 ( x a ) | ≤ | x a | (here we are using the well known fact that | sin y | ≤ | y | for every y R ). It follows that given any > 0, we have | f ( x ) f ( a ) | < whenever | x a | < min { , π } . Example 4.1.8. It is worthwhile to mention that the function f ( x ) = 1 if x is rational, 0 if x is irrational, is not continuous at x = a for any a R . In other words, f ( x ) is continuous nowhere. The proof is rather long and complicated. It depends on the well known fact that between any two real numbers, there are rational and irrational numbers.
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