Calculus/Multivariable Calculus/Limits and Continuity

Limits and continuity

Before we can look at derivatives of multivariate functions, we need to look at how limits work with functions of several variables first, just like in the single variable case.

If we have a function f : R^m → Rⁿ, we say that f(x) approaches b (in Rⁿ) as x approaches a (in R^m) if, for all positive ε, there is a corresponding positive number δ, |f(x)-b| < ε whenever |x-a| < δ, with x ≠ a.

This means that by making the difference between x and a smaller, we can make the difference between f(x) and b as small as we want.

If the above is true, we say

• f(x) has limit b at a
• $\lim _{\mathbf {x} \rightarrow \mathbf {a} }\mathbf {f} (\mathbf {x} )=\mathbf {b}$
• f(x) approaches b as x approaches a
• f(x) → b as x → a

These four statements are all equivalent.

Rules

Since this is an almost identical formulation of limits in the single variable case, many of the limit rules in the one variable case are the same as in the multivariate case.

For f and g, mapping R^m to Rⁿ, and h(x) a scalar function mapping R^m to R, with

• f(x) → b as x → a
• g(x) → c as x → a
• h(x) → H as x → a

then:

• $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} +\mathbf {g} )=\mathbf {b} +\mathbf {c}$
• $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(h\mathbf {f} )=H\mathbf {b}$

and consequently

• $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} \cdot \mathbf {g} )=\mathbf {b} \cdot \mathbf {c}$
• $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} \times \mathbf {g} )=\mathbf {b} \times \mathbf {c}$

when H≠0

• $\lim _{\mathbf {x} \rightarrow \mathbf {a} }({\mathbf {f} \over h})={\mathbf {b} \over H}$

Continuity

Again, we can use a similar definition to the one variable case to formulate a definition of continuity for multiple variables.

If f : R^m → Rⁿ, f is continuous at a point a in R^m if f(a) is defined and

$\lim _{\mathbf {x} \rightarrow \mathbf {a} }\mathbf {f} (\mathbf {x} )=\mathbf {f} (\mathbf {a} )$

Just as for functions of one dimension, if f, g are both continuous at p, f+g, λf (for a scalar λ), f·g, and f×g are continuous also. If φ : R^m → R is continus at p, φf, f/φ are too if φ is never zero.

From these facts we also have that if A is some matrix which is n×m in size, with x in R^m, a function f(x)=A x is continuous in that the function can be expanded in the form x₁a₁+...+x_ma_m, which can be easily verified from the points above.

If f : R^m → Rⁿ which is in the form f(x) = (f₁(x),...,f_n(x) is continuous if and only if each of its component functions are a polynomial or rational function, whenever they are defined.

Finally, if f is continuous at p, g is continuous at f(p), g(f(x)) is continuous at p.

Special note about limits

It is important to note that we can approach a point in more than one direction, and thus, the direction that we approach that point counts in our evaluation of the limit. It may be the case that a limit may exist moving in one direction, but not in another.