BAS CAL | Continuity

Continuity at a Number

A function is said to be continuous at number a if

1. f\left(a\right) exists    

2. \lim_{x\to a}f\left(x\right) exists, and

3. \lim_{x\to a}f\left(x\right)=f\left(a\right)

Types of Discontinuities

Removable

• the presence of a hole in the graph of the function

  • a hole happens when the direct substitution leads to the indeterminate form

• can be removed by redefining the function at the x-value of the hole and making f(a) equal to another number

Essential

• occurs when a function stops at one point and jumps to another

Infinite

• occurs when at least one of the two limits (left-hand and right-hand) is infinite

• also known as the asymptotic discontinuity, as it occurs when there is a asymptote at a

Continuity on a Closed Interval

A function is said to be continuous on a closed interval [a, b] if

1. It is continuous on the open interval (a, b).

2. It is continuous from the right of a

   • f\left(a\right) exists

   • \lim_{x\to a^{+}}f\left(x\right) exists, and

   • \lim_{x\to a^{+}}f\left(x\right)=f\left(a\right)

3. It is continuous from the left of b

   • f\left(b\right) exists

   • \lim_{x\to b^{+}}f\left(x\right) exists, and

   • \lim_{x\to b^{+}}f\left(x\right)=f\left(b\right)

Intermediate Value Theorem

• If a function is continuous over a closed interval [a, b], then for every value m between f(a) and f(b), there is a value c ∈ [a, b] such that f(c) = N.

Existence of Roots

• For a function that is continuous on a closed interval [a,b], suppose and f(a) and f(b) have different signs, then there exists c between a and b such that f(c) = 0.