MATH1011 - Lecture 3

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MATH1011 - Lecture 2
Week 1 Content Continued:

Polar Coordinates

• $(x,y)\to (0,0)$ corresponds $r\to 0$
  • Where $x^2+y^2=r^2$

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Skeleton Slides

Continuity of Functions

• A function is continuous if its graph has no graphs
• This is represented mathematically by: ${\lim }_{x\to c}f(x)=f(c)$
  i.e. f is continuous at c

Combining Continuous Functions

1. The functions $af(x)+bg(x)(a,b\in R)$ and $f(x)g(x)$ are continuous at $c$
2. If $g(c)\ne 0$, then $\frac{f(x)}{g(x)}$ is continuous at $c$
3. If $h(t)$ is a function of a single variable $t\in R$, defined on the range of $f$ and continuous at $f(c)$, then $h(f(x))$ is continuous at $c$
4. If $t\in I\subseteq R$, and $r(t)$ is a vector-valued function such that $r(t)\in D$ for $t\in I$ and $r(t_0)=c$, and if $r$ is continuous at $t_0$, then the function $f(r(t))$ is continuous at $t_0$

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MATH1011 - Lecture 4
Annotated Slides