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Function of a real variable - Quiz
Contributed by: Downes
  • 1. What is the domain of a function of several real variables?
A) A subset of R
B) Always the set of all real numbers
C) A subset of complex numbers
D) A set of integers
  • 2. What is the restriction of a function f to a subset Y denoted as?
A) fY
B) f(Y)
C) f|Y
D) Y(f)
  • 3. Why is it not always necessary to explicitly define the domain of a function?
A) Because the text states domains are irrelevant
B) Because domains are always the entire set of real numbers
C) Because it is sometimes possible to enlarge the domain naturally
D) Because functions do not require domains
  • 4. What is the domain of the constant function (x) ↦ r?
A) It depends on r
B) It is defined only for x > 0
C) It is everywhere defined
D) It has no domain
  • 5. When can functions f and g be added or multiplied?
A) If they are continuous
B) If they are both everywhere defined
C) If their domains are identical
D) If the intersection of their domains contains an open subset of R
  • 6. What structure do functions defined in a neighborhood of a point form?
A) Fields
B) Commutative algebras over the reals
C) Vector spaces without multiplication
D) Groups only
  • 7. Why are the algebras of functions not fields?
A) Because the reciprocal 1/f is not always defined
B) Because they lack an additive identity
C) Because they are not commutative
D) Because addition is not defined
  • 8. What is the distance function d(x, y) defined as?
A) sqrt(x2 + y2)
B) |x - y|
C) x - y
D) |x + y|
  • 9. When is a function f considered continuous at an interior point a?
A) If f(a) is zero
B) If the derivative exists at a
C) If for every ε > 0 there is a δ > 0 such that |f(x) - f(a)| < ε whenever d(x, a) < δ
D) If the limit as x approaches a is infinity
  • 10. What is the requirement for a function to be called continuous?
A) It must be a constant function
B) It must be continuous at every point of its domain
C) It must be defined everywhere
D) It must be differentiable
  • 11. What must be true for a limit L of a function f at a to exist?
A) f(a) must be equal to zero
B) The function must be constant in a neighborhood of a
C) The function must be defined at a
D) For every ε > 0 there exists δ > 0 such that |f(x) - L| < ε for all x in the domain where d(x, a) < δ
  • 12. If a is in the interior of the domain, what is the relationship between the limit and continuity?
A) The limit exists if and only if the function is continuous at a
B) The limit exists only if the function is constant
C) The limit always exists
D) The limit exists only if the function is differentiable
  • 13. What can be done when a is in the boundary of the domain and the limit exists?
A) Change the function into a constant function
B) Ignore the point a
C) Extend the domain of f to a by continuity
D) Define the derivative at a
  • 14. In older texts, the theory of functions of a real variable is synonymous with what field?
A) Topology
B) Linear algebra
C) Real analysis
D) Complex analysis
  • 15. What is the image of a function of a real variable called?
A) A domain
B) A matrix
C) A curve
D) A scalar
  • 16. When a function defines a curve, what is it called?
A) A vector field
B) A geometric identity
C) A linear projection
D) A parametric equation
  • 17. What structural property is often assumed for the codomain of a function of a real variable?
A) Boolean algebra
B) Integer lattice
C) Finite field
D) R-vector space
  • 18. Which of these is mentioned as a possible R-algebra for the codomain?
A) Empty set
B) Natural numbers
C) Prime numbers
D) Quaternions
  • 19. Is the sine function differentiable at every point of its domain?
A) No, only at positive points
B) Yes
C) No, only at negative points
D) No, it is not differentiable
  • 20. Is the absolute value function differentiable at zero?
A) No
B) Yes
C) Only for negative values
D) Only for positive values
  • 21. The tangent function is not defined at what points?
A) Pi/2 + k*pi
B) k*pi
C) Pi + k*pi
D) 2*pi + k*pi
  • 22. What technique is used for evaluating a mixture of integrals and derivatives?
A) Differentiation under the integral sign
B) Implicit differentiation
C) Taylor expansion
D) Integration by parts
  • 23. How many functions r_i(t) are required to describe a curve in Rn?
A) 1
B) n
C) n+1
D) n-1
  • 24. What is the common variable for all r_i(t) functions in the formulation?
A) t
B) x
C) n
D) r
  • 25. In the expression r(t) = [r1(t), r2(t), ..., r_n(t)], what does the resulting object represent?
A) A single real number
B) A matrix of size n by n
C) A parametrized n-tuple
D) A scalar field
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