Questions on the concepts of continuity and continuous functions in calculus are presented along with their answers. These questions have been designed to help you gain deep understanding of the concept of continuity.
Questions with Solutions
True or False. If a function f is not defined at x = a then it is not continuous at x = a. Answer :
True. See definition of continuous functions.
True or False. If f is a function such that
lim f(x) as x --> a
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. Answer :
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.