PureMath.pdf

# The properties thus defined are far from exhausting

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The properties thus defined are far from exhausting those which are possessed by the curve as pictured by the eye of common sense. This pic- ture of a curve is a generalisation from particular curves such as straight lines and circles. But they are the simplest and most fundamental proper- ties: and the graph of any function which has these properties would, so far as drawing it is practically possible, satisfy our geometrical feeling of what a continuous curve should be. We therefore select these properties as embodying the mathematical notion of continuity. We are thus led to the following Definition. The function φ ( x ) is said to be continuous for x = ξ if it tends to a limit as x tends to ξ from either side, and each of these limits

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[V : 99] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 212 is equal to φ ( ξ ) . We can now define continuity throughout an interval . The function φ ( x ) is said to be continuous throughout a certain interval of values of x if it is continuous for all values of x in that interval. It is said to be continuous everywhere if it is continuous for every value of x . Thus [ x ] is continuous in the interval [ δ, 1 - δ ], where δ is any positive number less than 1 2 ; and 1 and x are continuous everywhere. If we recur to the definitions of a limit we see that our definition is equivalent to ‘ φ ( x ) is continuous for x = ξ if, given , we can choose δ ( ) so that | φ ( x ) - φ ( ξ ) | < if 0 5 | x - ξ | 5 δ ( )’. We have often to consider functions defined only in an interval [ a, b ]. In this case it is convenient to make a slight and obvious change in our definition of continuity in so far as it concerns the particular points a and b . We shall then say that φ ( x ) is continuous for x = a if φ ( a + 0) exists and is equal to φ ( a ), and for x = b if φ ( b - 0) exists and is equal to φ ( b ). 99. The definition of continuity given in the last section may be illus- trated geometrically as follows. Draw the two horizontal lines y = φ ( ξ ) - and y = φ ( ξ ) + . Then | φ ( x ) - φ ( ξ ) | < expresses the fact that the point on the curve corresponding to x lies between these two lines. Similarly X Y 0 P ξ - δ ξ + δ y = φ ( ξ ) - ǫ y = φ ( ξ ) + ǫ Fig. 30. | x - ξ | 5 δ expresses the fact that x lies in the interval [ ξ - δ, ξ + δ ]. Thus
[V : 99] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 213 our definition asserts that if we draw two such horizontal lines, no matter how close together, we can always cut off a vertical strip of the plane by two vertical lines in such a way that all that part of the curve which is con- tained in the strip lies between the two horizontal lines. This is evidently true of the curve C ( Fig. 29 ), whatever value ξ may have. We shall now discuss the continuity of some special types of functions. Some of the results which follow were (as we pointed out at the time) tacitly assumed in Ch. II .

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