MTH2010 Chap.2 Functions of Several Variables, Limits and Continuity
Functions of Several Variables, Limits and Continuity
This chapter covers the shift from functions of one variable to functions of two or three. A function of two variables has a domain in the plane, a range of output values and a graph that is a surface in space.
Because surfaces are hard to draw, the unit relies on level curves, the sets where the function equals a constant, and a contour map of several level curves is often the quickest way to read off the range and the shape of a function. In three variables the corresponding objects are level surfaces. The second half of the chapter is about limits.
In two variables a point can be approached along infinitely many paths, and the definition of a limit requires every approach to give the same value. This produces the central asymmetry of the topic: one path along which the limit fails, or two paths with different limits, proves that the limit does not exist, while no number of agreeing paths can prove that it does.
To prove existence, the seminar slides offer the limit laws for continuous functions, the squeeze theorem, and the epsilon-delta definition. Continuity at a point means the limit exists and equals the function value, and polynomials, together with sums, products, quotients and compositions of continuous functions, are continuous wherever they are defined.
Exam questions pair a limit that can be squeezed to zero with one whose axis limits disagree.
What this chapter covers
- 01
Domain and range: inequalities for logarithms and roots, then range from level sets
- 02
Level curves and contour maps: every case of f = c, including a point or the empty set
- 03
Graphs versus level sets: why spheres need a level-set description
- 04
Two-variable limits: the definition requires agreement along every path
- 05
Disproving a limit: two paths, or a curved path when lines all agree
- 06
Proving a limit: limit laws, the squeeze theorem, or epsilon-delta
- 07
Polar substitution: bounds must hold uniformly in the angle
- 08
Continuity and extension: when a value at a point can repair a function
Worked example · free
Domain, range and level curves of a root function
- 1The root needs 16 - x² - 4y² ≥ 0, so the domain is the closed elliptical region x² + 4y² ≤ 16.
- 1The expression under the root runs from 0 on the boundary to 16 at the origin, so the range is 0 ≤ f ≤ 4.
- 1For 0 ≤ c < 4, f = c gives x² + 4y² = 16 - c², an ellipse that shrinks as c grows.
- 1When c = 4 only the origin satisfies the equation; for c outside [0, 4] it is empty.
Key terms
- Domain
- The set of input points at which a function's formula is defined.
- Level surface
- The set of points in space where a function of three variables equals a fixed constant.
- Squeeze theorem
- A rule stating that a function trapped between two others with a common limit shares that limit.
- Continuous function
- A function whose limit at each point of its domain equals its value there.
Functions of Several Variables, Limits and Continuity FAQ
Why are straight-line paths not enough to show a limit exists?
Some functions agree along every straight line through a point yet behave differently along a curve, such as a parabola. Agreement along any collection of paths is never a proof; a bound valid for all nearby points is needed.
How do I find the range of a function of two variables?
Solve f = c for a general constant and decide for which values the level set is non-empty. Those values of c form the range, and listing the cases also gives the level curves the question usually asks for.
When should I use polar coordinates for a limit at the origin?
Use them when the expression simplifies into a power of r times a bounded function of the angle. The bound must not depend on the angle, because the angle can change as r shrinks along a curved path.
Can redefining a function at one point make it continuous?
Only if the limit exists there. If two paths give different limits, no single value can match both, so the function cannot be extended continuously, and the answer should say so explicitly.
Exam move
Start with level curves, because they also answer most range questions: practise listing every case of f = c and sketching three or four contours. For limits, keep two separate checklists. To disprove, test the axes, then lines y = mx, then a curve that balances the degrees. To prove, look for a bound of the form |f - L| ≤ g with g tending to zero, using the inequality x² ≤ x² + y² or the boundedness of sine.
Rework one squeeze argument and one two-path argument from the practice mid-semester papers every few days until the first line comes without hesitation, and always end with a sentence naming the method you used.
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