CM12006 Continuity

What does continuous from the right mean? [2]

What does continuous from the left mean? [2]

What is continuous on an interval? [1]

Continuous at every number in the interval

What is continuous at an endpoint? [2]

If the interval is [a, b]

Continuous at a means it must be continuous from the right

f(a) should be defined and limit equals f(a)

How do you prove a function f is continuous on an interval e.g. [2, ∞)? [3]

Show its continuous at every number a > 2

Check continuity at endpoint a = 2

What is domain of continuity? [1]

Set of all numbers at which a function f is continuous

What does it mean by continuity of a function is preserved under basic mathematical operations? [3]

If f and g are continuous at a and c is a constant,

Then the following functions are also continuous at a

e.g. f + g, f - g, cf, fg, f / g if g(a) ≠ 0

What is continuity of composite functions? [2]

If g is continuous at a and f is continuous at g(a)

Then f(g(x)) is continuous at a

What is the gradient of the tangent line x → a? [2]

What is the equation of f’(a)? [2]

What does it mean for a function f to be differentiable at a? [1]

f’(a) exists

What does it mean for a function f to be differentiable on an open interval (a, b)? [1]

It is differentiable at every number in the interval

What are partial derivatives? [2]

When f is a function of 2 variables, x and y

There is a partial derivative for x and a partial derivative for y

How are partial derivatives denoted? [2]

f_x(x, y) and f_y(x, y)

How do you calculate f_x(x, y)? [1]

Just differentiate with respect to x, treat y terms as constants