HT2026 Continuous Maths notes

Definition 1.1: Let .

is continuous at if .

is continuous if it is continuous at every .

Definition 1.2: Let .

is continuous at if .

is continuous if it is continuous at every .

Theorem 1.3: If and are continuous, so are:

• , when strictly positive
• , where strictly positive
• , where nonzero

Definition 1.4: Let .

is differentiable at if exists. Then is the derivative of at .

is differentiable if it is differentiable at every .

Remark: If a function is not continuous, it is not differentiable.

Definition 1.5: A secant is a line that connects two points on a surface.

A tangent can be thought of as a limit of secants as the distance between the two points tend to 0.

Remark: If and are differentiable, so are all the combinations that preserve continuity mentioned earlier, with the exception of and .

Remark: The derivative is a linear operator.

Theorem 1.6 (Quotient rule):

Theorem 1.7 (Derived quotient rule):

Proof: TODO (problem sheet 0)

Theorem 1.8 (Taylor's Theorem (univariate)): For ,

for some .

The Taylor polynomial (of degree ) is also written , and the error term , or Lagrange remainder term is denoted by .

So, we can write for any so long as the first derivatives of all exist and are continuous on .

For small , .

Remark: is the unique -order polynomial that agrees with as to its first derivatives at .

Remark:Taylor’s Theorem is not the same as a Taylor series; the Taylor series is an infinite sum and does not always converge to the correct answer. (If tends to 0 for large and small enough , the series does converge and that function is analytic ).

Definition 1.9: Let .

is the partial derivative of with respect to , as if were constant.

means differentiate first with respect to , then . In general this is not equal to , but often it is.

Definition 1.10: Let . Then the vector derivative is

It turns out that this is equal to

Remark: Some standard derivatives for vectors that are analogous to scalars:

Remark: Similarly to scalar derivatives, the vector derivative is linear and has a product and quotient rule.

The chain rule for is the same as for scalars.

For is a bit different - see TODO