2.3 - Continuity

Testing for continuity at a point

Interior Points

A function $y=f(x)$ is continuous at an interior point $c$ of its domain if ${\displaystyle \underset{x\to c}{lim}=f(c)}$.

Endpoints

A function $y=f(x)$ is continuous at a left endpoint $a$ or a right endpoint $b$ of its domain if ${\displaystyle \underset{x\to a^{+}}{lim}=f(a)}$ or ${\displaystyle \underset{x\to b^{-}}{lim}=f(b)}$ respectively.

TLDR

Step 1: Does ${\displaystyle \underset{x\to c}{lim}f(x)}$ exist?
(are the left and right hand limits equal?)
Step 2: Does $f(c)$ exist?
Step 3: Do the limit and function value equal one another?
For these types of questions it’s important to have some kind of limit value statement in your answer.

Types of (dis)continuity

• Jump continuity
  • e.g. piecewise functions
• Removable discontinuity
  • i.e. a hole
• Infinite discontinuity
  • e.g. a vertical asymptote
• Oscillating discontinuity (very rare)
  • e.g. $y=\sin \left(\frac{1}{x}\right)$

Removing a discontinuity (“filling in holes”)

Essentially just cancel terms that make the denominator zero and then evaluate the function at that point.

Extrema

• Just maximums and minimums of a function
• You can have local and absolute extrema
  • Local extrema are local to a certain sub-interval
  • Absolute extrema are the absolute greatest/lowest in a given interval

Continuity and extrema

• If a function $y=f(x)$ is continuous in $[a,b$], then $f(x)$ has both an absolute maximum and minimum on that interval.
  • We could say “$f(x)$ has a max of $6$ at $x=3$”
• In an open interval (e.g. $(1,1)$) the absolute maximum/minimum can’t be the endpoints.

The Intermediate Value Theorem