Functions of Several Variables, Limits and Continuity: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 3, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A rule assigning exactly one real output to each ordered pair drawn from a domain in the plane. Its graph lives in three dimensions as a surface over that domain.
Domain of a multivariable function
The set of input points where the formula makes sense, typically restricted by division by zero, even roots of negatives and logarithms of nonpositive numbers.
Range of a multivariable function
The set of output values actually attained as the input ranges over the domain, always a subset of the real numbers for a scalar-valued function.
Level curve
The set of domain points where a function of two variables takes a fixed value. Drawing several of them is the contour map view of the surface.
Contour map
A family of level curves labelled by their values. Closely spaced contours indicate steep terrain, and the spacing pattern reveals ridges, valleys and saddles before any derivative is computed.
Level surface
The set of points in space where a function of three variables equals a constant. Since the graph would need four dimensions, level surfaces are the practical visualisation.
Function of three variables
A rule assigning one real number to each ordered triple in a spatial domain, such as a temperature or density field filling a region.
Open disk
The set of plane points strictly within a fixed positive distance of a centre. Open disks are the neighbourhoods used to state limits in two variables.
Interior point
A point of a set surrounded entirely by an open disk contained in the set. A set is open when every one of its points is interior.
Boundary point
A point every neighbourhood of which meets both the set and its complement. Boundary points may or may not belong to the set itself.
Closed set in the plane
A set containing all of its boundary points. Closedness is one of the two hypotheses in the extreme value theorem for several variables.
Bounded set
A set contained inside some disk of finite radius. Without boundedness a continuous function on a closed set may fail to attain a maximum.
Limit of a function of two variables
The single value approached as the input point nears a target from every possible direction and along every possible path within the domain.
Two-path test
A method for proving a limit does not exist: find two approach paths giving different limiting values. It can disprove existence but can never prove it.
Approach along the axes
Substituting one variable equal to zero to test a limit at the origin. Agreement along both axes is far from sufficient, since infinitely many other paths remain.
Approach along a line through the origin
Substituting one variable as a constant multiple of the other. A limit depending on that constant proves nonexistence immediately.
Approach along a parabolic path
Substituting one variable as a multiple of the square of the other, needed when all straight lines give the same value but the limit still fails to exist.
Polar coordinate limit method
Rewriting a limit at the origin in terms of radius and angle. If the expression is bounded by a function of radius alone that tends to zero, the limit exists and equals zero.
Squeeze argument in two variables
Bounding the absolute difference between a function and a candidate limit by an expression that tends to zero. This is one of the few techniques that genuinely proves existence.
Continuity at a point in several variables
Requires the function to be defined there, the limit to exist there, and the two values to agree. Failure of any one of the three breaks continuity.
Polynomial function of several variables
A finite sum of terms, each a constant times nonnegative integer powers of the variables. Such functions are continuous everywhere in space.
Rational function of several variables
A quotient of two polynomials in several variables, continuous everywhere its denominator is nonzero. The zero set of the denominator is exactly where trouble can occur.
Composition rule for continuity
A continuous function of one variable applied to a continuous multivariable function yields a continuous result on the appropriate domain, which classifies most textbook functions at a glance.
Removable versus essential discontinuity in two variables
A discontinuity is removable when the limit exists but the assigned value differs; it is essential when the limit itself fails to exist, as in path-dependent quotients.
Graph of a function of two variables
The set of points in space whose height coordinate equals the function value at the base point. It is a surface, and it passes the vertical line test by construction.