lim f(x) as x --> a
True or False. If f is a function such that
does not exist then f is not continuous.
True. For a function to be continuous at x = a, lim f(x) as x approaches a must be equal to f(a) and obviously the limit must exist and f(x) must be defined at x = a.
True or False. All polynomial functions are continuous.
True. It is a theorem on continuity of polynomials.
If functions f(x) and g(x) are continuous everywhere then
(A) (f / g)(x) is also continuous everywhere.
(B) (f / g)(x) is also continuous everywhere except at the zeros of g(x).
(C) more information is needed to answer this question
(B) . Students tends to forget about the zeros of g(x) for which (f / g)(x) is undefined.
If functions f(x) and g(x) are continuous everywhere and f(1) = 2, f(3) = -4, f(4) = 8, g(0) = 4, g(3) = -6 and g(7) = 0 then lim (f + g)(x) as x approaches 3 is equal to
(C) cannot find a value for the above limit since only values of the functions are given.
(A) . lim (f + g)(x) = lim f(x) + lim g(x) and since the two functions are continuous then the limits are equal to the values of the functions at x = 3. Hence (as x approaches 3) lim (f + g)(x) = lim f(x) + lim g(x) = f(3) + g(3) = -10.
The following statement is true.
If f(x) = sin x, then f is a continuous function.
Which of the following is also true?
(A) if f(x) is not equal to sin x, then f is not continuous.
(B) if f is not a continuous function, then f(x) is not equal to sin x.
(C) if f is continuous, then f(x) = sin x
(B) . This is the contrapositive of the given statement.
In logic, the contrapositive of the statement "if p then q" is "if not q then not p".
It is easy to find examples that show that (A) and (C) are not (always) correct.
True or False. If f(x) is continuous everywhere, then |f(x)| is continous everywhere.
True. See the theorem on the composition of continuous functions: here f(x) and | x | are continuous everywhere.
True or False. If f(x) is continuous everywhere, then square root [ f(x) ] is continuous everywhere.
False. square root [ f(x) ] is continuous for f(x) nonnegative.
True or False. If the composition f o g is not continuous at x = a, then either g is not continuous at x = a or f is not continuous at g(a).
True. Contrapositive of the theorem on composition of continuous functions.
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