2.5 HW Problems

Table of Contents:

Continuity at a point (video) | Khan Academy

• Any function $y=f(x)$ whose graph can be sketched over its domain in one unbroken motion is an example of a continuous function.

Continuity at a Point

At which numbers does the function f appear to be NOT continuous? Explain why. What occurs at other numbers in the domain?

• We see that the domain of the function is a closed interval at $[0,4]$.

• There are break at $x=1$, $x=2$, and $x=4$

• The break at $x=1$ is a “jump discontinuity”.

• The break at $x=2$ and $x=4$ is a “removable discontinuity”.

• We say a function is continuous at a number c if:

$$ \lim_{x\rightarrow c} f(x)=f(c) $$

• We say a function is continuous from the right if:

  $$ \lim_{x\rightarrow c^+} f(x)=f(c) $$

• We say a function is continuous from the left if:

  $$ \lim_{x\rightarrow c^-} f(x)=f(c) $$

• We say a function is continuous on [a,b] if it is continuous for all $c \epsilon (a,b)$ and continuous from the right at a and continuous from the left at b

  • “you can draw a line without picking up your pencil”

    

  <aside> 💡 Continuity Test: A function f(x) is continuous at a point $x = c$ if and only if it meets the following three conditions:

  1. $f(c)$ exists (c lies in the domain of f)
  2. $\lim_{x \rightarrow c} f(x)$ exists (f has a limit as $x \rightarrow c$)
  3. $\lim_{x \rightarrow c} f(x) = f(c)$ (the limit equals the function value) </aside>