# A First Course in Real Analysis, Second Edition by Murray H. Protter, Charles Bradfield Morrey

By Murray H. Protter, Charles Bradfield Morrey

Many adjustments were made during this moment version of A First direction in actual research. the main seen is the addition of many difficulties and the inclusion of solutions to many of the odd-numbered routines. The book's clarity has additionally been stronger by means of the extra explanation of some of the proofs, extra explanatory feedback, and clearer notation.

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**Example text**

A function f is continuous on the right at a if and only if a is in the domain off and f(x)-+ f(a) as x -+ a+. The function f is continuous on the left at a if and only if a is in the domain off and f(x)-+ f(a) as x-+ a-. If the domain of a function f is a finite interval, say a ~ x ~ b, then limits and continuity at the endpoints are of the one-sided variety. For example, the function f: x -+ x 2 - 3x + 5 defined on the interval 2 ~ x ~ 4 is continuous on the right at x = 2 and continuous on the left at x = 4.

Solution. If x :;l: 4, then )(Jx f(x) =

So long as we know which objects are in a given set and which are not. If S is a set and P is an object in it, we write P E S and say that P is an element of S or that P belongs to S. If S1 and S2 are two sets, their union, denoted by S1 u S2 , consists of all objects each of which is in at least one of the two sets. The intersection of S1 and S2 , denoted by S 1 n S2 , consists of all objects each of which is in both sets. 5), and S2 is the vertically shaded set, then S1 u S2 consists of the entire shaded area, and S1 n S2 consists of the doubly shaded area.