7 Functions, Limits and Continuity 7.1 Revision and Examples A - - PDF document

7 Functions, Limits and Continuity

7.1 Revision and Examples

A function f : A → B from a set A (the domain of f) to a set B (the co-domain of f) is a rule assigning to each a ∈ A a unique element f(a) ∈ B. We write a → f(a) to stress this. If S ⊆ A is a subset then we define the restriction f|S of f to S to be the function f|S : S → B given by (f|S)(s) = f(s) for all s ∈ S. It is still the same rule but only applied to elements in S so we only change the domain of the function (which is part

• f its definition).

Given a function f : A → B, the image of S under f is the set f(S) = {f(s) | s ∈ S} = {b ∈ B | there exists s ∈ S such that f(s) = b} ⊆ B . The image of f is simply the image of the whole of the domain of f under f, i.e., f(A) (since the domain of our function is A).

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In G11ACF you encountered mainly functions f : [a, b] → R or g : R → R i.e. functions of a single (real) variable defined on a closed subinterval or the whole of R (i.e. with domain equal to [a, b] or R) taking values in R (i.e. co-domain equal to R). Gap to fill in In G11CAL you saw functions f : R2 → R or g : R3 → R

• r ones defined on subsets of R2 or R3 respectively

(functions of two or three variables). Gap to fill in

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We will consider functions functions of the type f : Rd → Rl

• r f : D → Rl,

where d, l are positive integers and D ⊆ Rd. In the sequel d and l denote fixed positive integers unless otherwise specified. Given f : R → Rl, f(x) has l coordinates, and so f is given by an l-tuple of real-valued functions in the following way: we may write f(x) = (f1(x), f2(x), . . . , fl(x)) for l real-valued functions f1, f2, . . . , fl (one for each coordinate in Rl). Gap to fill in

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In the same way, any function f : Rd → Rl or g : D → Rl is given by an l-tuple of real valued functions f1, f2, . . . , fl : Rd → R respectively g1, g2, . . . gl : D → R, where f(x) = (f1(x), f2(x), . . . , fl(x)) for x ∈ Rd, and g(y) = (g1(y), g2(y), . . . , gl(y)) for y ∈ D. You saw a large number of examples in G11ACF and G11CAL, often made up from polynomials, rational functions, exponential functions, logarithmic functions and trigonometric functions. Gap to fill in

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You should bear these examples in mind, and remind yourselves of their properties. We shall assume without further proof that these standard functions have the properties claimed in G11CAL. Gap to fill in See books for formal proofs of facts concerning the continuity and differentiability of these standard functions.

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7.2 Limits and Continuity

We recall the notions of limit and continuity for functions

• f one variable from G11ACF and G11CAL.

Let a ∈ R and let f be a real-valued function defined (at least) at all points of R \ {a} (but not necessarily at a). Then the limit limx→a f(x) exists and equals L ∈ R if, for every sequence (xn) ⊆ R \ {a} which converges to a, we have f(xn) → L as n → ∞. The notation lim

x→a f(x) = L

means that the limit exists and is equal to L. Note that f(a) was not necessarily defined above, and so was not involved when investigating the above limit. Now suppose that we have f : R → R. Then f is said to be continuous at a ∈ R if lim

x→a f(x) = f(a) . 6

Since we know what convergence of sequences in Rd and Rl means, we can generalize the definition of limits to functions f : Rd → Rl. Definition 7.2.1 Let a ∈ Rd,let q ∈ Rl, and let f be a function taking values in Rl which is defined (at least) at all points of Rd \ {a}. We say that the limit limx→a f(x) of f at a exists and equals q if, for every sequence (xn) ⊆ Rd \ {a} which converges to a, we have f(xn) → q (in Rl) as n → ∞. In this case, we may also write f(x) → q as x → a. If there is no such q in Rl, then the limit above does not exist. As before, if we write either limx→a f(x) = q or f(x) → q as x → a, we mean that the limit exists and is equal to q.

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Now suppose that we have a function f : Rd → Rl, and let a ∈ Rd. Then f is said to be continuous at a if lim

x→a f(x) = f(a) .

Otherwise we say that f is discontinuous at a: this means that either the limit above does not exist or it does exist, but is not equal to f(a). Note that f : Rd → Rl is continuous at a if and

• nly if the following condition holds:

for every se- quence (xn) ⊆ Rd which converges to a, we have f(xn) → f(a) as n → ∞. This time we do not insist on xn = a. [Exercise: check the details.]

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We now wish to consider functions defined on a non-empty subset D of Rd. However, when discussing limits, we will meet an obstacle when it comes to ‘isolated’ points. A point a of D is isolated if there is an r > 0 such that a is the only point in D ∩ Br(a): informally, there are no

• ther points of D near to a.

Gap to fill in

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Let D be a non-empty subset of Rd, and suppose that we have a function f : D → Rl or f : D \ {a} → Rl. We define limits and continuity in a similar way to above. For such f, and a ∈ D, we say that limx→a f(x) exists and equals q ∈ Rl if, for every sequence (xn) ⊆ D \ {a} which converges to a, we have f(xn) → q as n → ∞. If there is no such q in Rl, then the limit above does not exist. Again, if we write either limx→a f(x) = q or f(x) → q as x → a, we mean that the limit exists and is equal to q. The function f : D → Rl is said to be continuous at a ∈ D if lim

x→a f(x) = f(a) .

Otherwise, we say that f is discontinuous at a. There is one problem with this definition of limit. If a is an isolated point of D then there are no such sequences (xn)! [Exercise. What do the above definitions mean if a is an isolated point?]

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Again, we may reformulate the definition of continuity in terms of sequences in the following way. This will be one

• f the most useful forms of the definition of

continuity for us. The function f is continuous at a if and only if the following condition holds: for every sequence (xn) ⊆ D which converges to a, we have f(xn) → f(a) as n → ∞. Here some or all of the xn may be equal to a. Note that this time we do not need to worry about isolated points, as we always have at least the constant sequence a, a, a, . . . available. [Exercise. What does this condition mean if a is an isolated point?]

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Definition 7.2.2 A function f : D → Rl is said to be continuous if it is continuous at every a ∈ D. Otherwise the function f is said to be discontinuous. Thus a function is discontinuous if there is at least one point of its domain at which it is discontinuous. We shall use without further proof the fact that the standard functions met in G11CAL are continuous on their domains of definition. In particular, ‘coordinate projections’ are continuous. Co-ordinate projections are maps such as the map from R2 to R given by (x, y) → x or (x, y) → y. More generally, for D ⊆ Rd and 1 ≤ i ≤ d, the ith coordinate projection pi : D → R is defined by pi((x1, x2, . . . , xd)) = xi. Gap to fill in

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Warning! It does not make sense to ask whether or not a function is continuous or discontinuous at a point where it is undefined! These questions only make sense at points of the domain of f. Gap to fill in

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7.3 Meaning of continuity for functions of several variables

We now discuss some of the many different ways there are to ‘approach’ a point in R2. This can be helpful when we want to establish that a function is discontinuous. Establishing that a function is continuous can be trickier! We will return to this issue in the next chapter. Example Let f : R2 → R be defined by f(x, y) =   

2xy x2+y2

if x2 + y2 > 0,

• therwise.

Does the limit lim(x,y)→(0,0) f(x, y) exist? Is f continuous at (0, 0)? Gap to fill in

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Notice that f defined above is continuous if we fix x and regard only y as a variable or fix y and regard only x as variable. This continuity in each variable of a function of two (or more variables) is called separate continuity whereas the type of continuity defined in 7.2.1 and 7.2.2 is sometimes called joint continuity. By the above example a function of several variables may well be separately continuous without being (jointly) continuous.

• Example. Determine whether or not the following

function g : R2 → R is continuous at the point (0, 0): g(x, y) =   

xy2 x2+y4

if x2 + y2 > 0,

• therwise.

Gap to fill in

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When investigating the continuity of specific examples, the following is very useful. The proof of this result is NEB, but the statement and applications are examinable. Let D ⊆ Rd and f : D → Rl be a function. As usual, write f(x) = (f1(x), f2(x), . . . , fl(x)) for x ∈ D, where f1, f2, . . . , fl are functions from D to R. Lemma 7.3.1 The function f : D → Rl is continuous at a ∈ D if and only if all the real valued functions f1, f2, .... fl are continuous at a. Examples. Gap to fill in

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