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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) A subset of complex numbers
C) Always the set of all real numbers
D) A set of integers
  • 2. What is the restriction of a function f to a subset Y denoted as?
A) f|Y
B) Y(f)
C) fY
D) f(Y)
  • 3. Why is it not always necessary to explicitly define the domain of a function?
A) Because it is sometimes possible to enlarge the domain naturally
B) Because the text states domains are irrelevant
C) Because functions do not require domains
D) Because domains are always the entire set of real numbers
  • 4. What is the domain of the constant function (x) ↦ r?
A) It depends on r
B) It is everywhere defined
C) It has no domain
D) It is defined only for x > 0
  • 5. When can functions f and g be added or multiplied?
A) If they are both everywhere defined
B) If the intersection of their domains contains an open subset of R
C) If they are continuous
D) If their domains are identical
  • 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 addition is not defined
B) Because the reciprocal 1/f is not always defined
C) Because they are not commutative
D) Because they lack an additive identity
  • 8. What is the distance function d(x, y) defined as?
A) |x + y|
B) x - y
C) |x - y|
D) sqrt(x2 + y2)
  • 9. When is a function f considered continuous at an interior point a?
A) If the limit as x approaches a is infinity
B) If the derivative exists at a
C) If f(a) is zero
D) If for every ε > 0 there is a δ > 0 such that |f(x) - f(a)| < ε whenever d(x, a) < δ
  • 10. What is the requirement for a function to be called continuous?
A) It must be defined everywhere
B) It must be a constant function
C) It must be differentiable
D) It must be continuous at every point of its domain
  • 11. What must be true for a limit L of a function f at a to exist?
A) 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
B) The function must be defined at a
C) The function must be constant in a neighborhood of a
D) f(a) must be equal to zero
  • 12. If a is in the interior of the domain, what is the relationship between the limit and continuity?
A) The limit exists only if the function is constant
B) The limit always exists
C) The limit exists if and only if the function is continuous at a
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) Ignore the point a
B) Extend the domain of f to a by continuity
C) Define the derivative at a
D) Change the function into a constant function
  • 14. In older texts, the theory of functions of a real variable is synonymous with what field?
A) Complex analysis
B) Real analysis
C) Topology
D) Linear algebra
  • 15. What is the image of a function of a real variable called?
A) A scalar
B) A domain
C) A matrix
D) A curve
  • 16. When a function defines a curve, what is it called?
A) A linear projection
B) A parametric equation
C) A geometric identity
D) A vector field
  • 17. What structural property is often assumed for the codomain of a function of a real variable?
A) R-vector space
B) Boolean algebra
C) Finite field
D) Integer lattice
  • 18. Which of these is mentioned as a possible R-algebra for the codomain?
A) Natural numbers
B) Prime numbers
C) Empty set
D) Quaternions
  • 19. Is the sine function differentiable at every point of its domain?
A) No, it is not differentiable
B) Yes
C) No, only at positive points
D) No, only at negative points
  • 20. Is the absolute value function differentiable at zero?
A) Yes
B) No
C) Only for positive values
D) Only for negative values
  • 21. The tangent function is not defined at what points?
A) Pi + k*pi
B) Pi/2 + k*pi
C) k*pi
D) 2*pi + k*pi
  • 22. What technique is used for evaluating a mixture of integrals and derivatives?
A) Integration by parts
B) Differentiation under the integral sign
C) Taylor expansion
D) Implicit differentiation
  • 23. How many functions r_i(t) are required to describe a curve in Rn?
A) n+1
B) n-1
C) n
D) 1
  • 24. What is the common variable for all r_i(t) functions in the formulation?
A) n
B) t
C) r
D) x
  • 25. In the expression r(t) = [r1(t), r2(t), ..., r_n(t)], what does the resulting object represent?
A) A parametrized n-tuple
B) A single real number
C) A scalar field
D) A matrix of size n by n
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