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 of 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 ). 1

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 : R 2 → R or g : R 3 → R or ones defined on subsets of R 2 or R 3 respectively (functions of two or three variables). Gap to fill in 2

We will consider functions functions of the type f : R d → R l or f : D → R l , where d, l are positive integers and D ⊆ R d . In the sequel d and l denote fixed positive integers unless otherwise specified. Given f : R → R l , 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 ) = ( f 1 ( x ) , f 2 ( x ) , . . . , f l ( x )) for l real-valued functions f 1 , f 2 , . . . , f l (one for each coordinate in R l ). Gap to fill in 3

In the same way, any function f : R d → R l or g : D → R l is given by an l -tuple of real valued functions f 1 , f 2 , . . . , f l : R d → R respectively g 1 , g 2 , . . . g l : D → R , where f ( x ) = ( f 1 ( x ) , f 2 ( x ) , . . . , f l ( x )) for x ∈ R d , and g ( y ) = ( g 1 ( y ) , g 2 ( y ) , . . . , g l ( 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 4

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. 5

7.2 Limits and Continuity We recall the notions of limit and continuity for functions of 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 lim x → a f ( x ) exists and equals L ∈ R if, for every sequence ( x n ) ⊆ R \ { a } which converges to a , we have f ( x n ) → L as n → ∞ . The notation x → a f ( x ) = L lim 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 x → a f ( x ) = f ( a ) . lim 6

Since we know what convergence of sequences in R d and R l means, we can generalize the definition of limits to functions f : R d → R l . Definition 7.2.1 Let a ∈ R d ,let q ∈ R l , and let f be a function taking values in R l which is defined (at least) at all points of R d \ { a } . We say that the limit lim x → a f ( x ) of f at a exists and equals q if, for every sequence ( x n ) ⊆ R d \ { a } which converges to a , we have f ( x n ) → q (in R l ) as n → ∞ . In this case, we may also write f ( x ) → q as x → a . If there is no such q in R l , then the limit above does not exist . As before, if we write either lim x → a f ( x ) = q or f ( x ) → q as x → a , we mean that the limit exists and is equal to q . 7

Now suppose that we have a function f : R d → R l , and let a ∈ R d . Then f is said to be continuous at a if x → a f ( x ) = f ( a ) . lim 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 : R d → R l is continuous at a if and only if the following condition holds: for every se- quence ( x n ) ⊆ R d which converges to a , we have f ( x n ) → f ( a ) as n → ∞ . This time we do not insist on x n � = a . [ Exercise: check the details.] 8

We now wish to consider functions defined on a non-empty subset D of R d . 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 ∩ B r ( a ) : informally, there are no other points of D near to a . Gap to fill in 9

Let D be a non-empty subset of R d , and suppose that we have a function f : D → R l or f : D \ { a } → R l . We define limits and continuity in a similar way to above. For such f , and a ∈ D , we say that lim x → a f ( x ) exists and equals q ∈ R l if, for every sequence ( x n ) ⊆ D \ { a } which converges to a , we have f ( x n ) → q as n → ∞ . If there is no such q in R l , then the limit above does not exist . Again, if we write either lim x → 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 → R l is said to be continuous at a ∈ D if x → a f ( x ) = f ( a ) . lim 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 ( x n ) ! [ Exercise. What do the above definitions mean if a is an isolated point?] 10

Again, we may reformulate the definition of continuity in terms of sequences in the following way. This will be one of 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 ( x n ) ⊆ D which converges to a , we have f ( x n ) → f ( a ) as n → ∞ . Here some or all of the x n 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?] 11

Definition 7.2.2 A function f : D → R l 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 R 2 to R given by ( x, y ) �→ x or ( x, y ) �→ y . More generally, for D ⊆ R d and 1 ≤ i ≤ d , the i th coordinate projection p i : D �→ R is defined by p i (( x 1 , x 2 , . . . , x d )) = x i . Gap to fill in 12

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 13

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 R 2 . 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 : R 2 → R be defined by  if x 2 + y 2 > 0 , 2 xy  x 2 + y 2 f ( x, y ) = 0 otherwise.  Does the limit lim ( x,y ) → (0 , 0) f ( x, y ) exist? Is f continuous at (0 , 0) ? Gap to fill in 14

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 : R 2 → R is continuous at the point (0 , 0) :  if x 2 + y 2 > 0 , xy 2  x 2 + y 4 g ( x, y ) = 0 otherwise.  Gap to fill in 15

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 ⊆ R d and f : D → R l be a function. As usual, write f ( x ) = ( f 1 ( x ) , f 2 ( x ) , . . . , f l ( x )) for x ∈ D , where f 1 , f 2 , . . . , f l are functions from D to R . Lemma 7.3.1 The function f : D → R l is continuous at a ∈ D if and only if all the real valued functions f 1 , f 2 , .... f l are continuous at a . Examples. Gap to fill in 16