275Continuity and Derivative

f is continuous at x= a if

What are the three types of discontinuity

The three types of discontinuity are point discontinuity, jump discontinuity, and infinite discontinuity.

What is point/removable discontinuity

hole in function

two sides agree with each other, but not the final function.

what is jump discontinuity

The function has a sudden change in value at a certain point, where the left-hand limit and right-hand limit exist but are not equal to each other.

Infinite discontinuity

Occurs when a function approaches infinity at a certain point, leading to unbounded behavior. This type of discontinuity typically results from division by zero or vertical asymptotes.

When is a function differentiable

What is the theorem of continuity

what are the properties of f and g for continuity

how to solve a continuity/ differentiability equation?

determine which sides go from left or right depending on domain. set equal to each other and discover what variables have to be for this to be true. make sure to evaluate at f(x) itself too. To find if its differentiable, plug these values in to the DERIVATIVES of the queations the equation and see what the variables equal.

special case

x=2 is not a problem in the first equation because the domain is x<1

Definition of derivative

Definition of tangent line

if x approaches negative 1

coming form the left

if x approaches pos 1

coming fomr the right