Continuity

• $f$ is continuous at $a$ iff $\underset{x\to a}{lim}f(x)=f(a)$
  • this means $f(a)$ is defined
  • and $\underset{x\to a}{lim}f(x)$ exists
  • cusps are continuous, but not jumps, holes, or asymptotes
• If $f$ and $g$ are continuous, then the following are also continuous
  • $f\pm g$
  • $fg$
  • $f\circ g$
  • $\frac{f}{g},g\ne 0$
• If you have a piecewise function with some variable, you can make the function continuous at some point by equating the limits on both sides of the discontinuity
  • You must make sure restrictions are satisfied
• $f(x)$ being differentiable at $x=a⟹f(x)$ is cts at $x=a$

Intermediate Value Theorem

• If $f(x)$ is continuous on $[a,b]$, and $N$ is a number between $f(a)$ and $f(b)$, then $\mathrm{\exists }c\in (a,b)$, such that $f(c)=N$
  • There may be multiple values of $c$
• You can use this theorem with a positive $a$ and a negative $b$ to find a root, or to show two functions are equal at some point
  • Take the difference of the two functions, and show that one side of an interval is less than 0, and one side is more (also prove cts)
  • If you have a cts function $g$ on interval $[0,2L]$, and $g(0)=g(2L)$, then $\mathrm{\exists }x\in [0,L]$ such that $g(x)=g(x+L)$
    • If proving this with an odd function, also show the case where $g(x)=0=g(x+L)$

Extreme Value Theorem

• If $f$ is continuous on a closed interval, there's a max and min value in there
• Could be either end