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15.2. THEOREM. Let a be a point in the domain D of the function f. The following statements are equivalent: (a) f is continuous at a. (b) If e is any positive real number, there exists a positive number 8(e) such that if x € D and x – al < ô(e), then |f(x) - f(a)| < e. (c) If (xn) is any sequence of elements of D which converges to a, then the sequence (f(xn)) converges to f (a).

15.2. THEOREM. Let a be a point in the domain D of the function f. The following statements are equivalent: (a) f is continuous at a. (b) If e is any positive real number, there exists a positive number 8(e) such that if x € D and x – al < ô(e), then |f(x) - f(a)| < e. (c) If (xn) is any sequence of elements of D which converges to a, then the sequence (f(xn)) converges to f (a).

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.CR: Chapter 3 Review
Problem 6CR
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15.L Solve based on the theorem in photo 2, please

15.L. We say that a function f on R to R is additive if it satisfies f(x + y) = f(x) + f(y) for all x, y continuous at any point of R. Show that a monotone additive function is con- tinuous at every point. R. Show that an additive function which is continuous at x = O is
Transcribed Image Text:15.L. We say that a function f on R to R is additive if it satisfies f(x + y) = f(x) + f(y) for all x, y continuous at any point of R. Show that a monotone additive function is con- tinuous at every point. R. Show that an additive function which is continuous at x = O is
15.2. THEOREM. Let a be a point in the domain D of the function f. The following statements are eguivalent: (a) f is continuous at a. (b) If e is any positive real number, there exists a positive number ô (e) such that if x € D and x – al < 8(e), then \f(x) – f(a)| < e. (c) If (x,) is any sequence of elements of D which converges to a, then the sequence (f (xn)) converges to f(a).
Transcribed Image Text:15.2. THEOREM. Let a be a point in the domain D of the function f. The following statements are eguivalent: (a) f is continuous at a. (b) If e is any positive real number, there exists a positive number ô (e) such that if x € D and x – al < 8(e), then \f(x) – f(a)| < e. (c) If (x,) is any sequence of elements of D which converges to a, then the sequence (f (xn)) converges to f(a).
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