Continuous Functions

We saw a few "nice" functions whose limits as $\mathrm{x} \rightarrow \mathrm{a}$ simply involved substituting $\mathrm{a}$ into the function: $\lim\limits_{x \rightarrow a} \mathrm{f}(\mathrm{x})=\mathrm{f}(\mathrm{a})$. Functions whose limits have this substitution property are called continuous functions, and they have a number of other useful properties and are very common in applications. We will examine what it means graphically for a function to be continuous or not continuous. Some properties of continuous functions will be given, and we will look at a few applications of these properties including a way to solve horrible equations such as $\sin (x)=\frac{2 x+1}{x-2}$.

Source: Dave Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-2.4-Continuous-Functions.pdf
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