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2.5  Defn:  $f$ is continuous at $a$ if $lim_{x \rightarrow a} f(x) = f(a)$

(i.e., if $lim_{x \rightarrow a} f(x) = f(lim_{x \rightarrow a} x) $

Examples:






Read left and right continuity

If $f$, $g$ continuous at $a$, $c \in {\cal R}$, then $f + g$, $fg$, $cf$, $f/g$ (if $g(a) \not= 0$) 
are continuous.

If $g$ continuous at $a$ and $f$ continuous at $g(a)$, then $f \circ g$ continuous at $a$.


Ex:  $lim_{x \rightarrow 0} {x^2 - e^{x^3} \over cos(x)} =$ 


Intermediate value theorem: Suppose $f$ continuous on $[a, b]$, $f(a) \not= f(b)$ and $n$ is between 
$f(a)$ and $f(b)$,  then there exists $c \in (a, b)$ such that $f(c) = N$.




Example:  Show that $x^2 - 7x + 1 $ has a root between 0 and 1.





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