3.1 Section 1.1

3.1.1 definition of limit of function $f(x)$

We say that the limit to $f(x)$ is $L$ as $x$ gets close to $c$, if for each number $\epsilon$ we can find another number $\delta$ such that $|f(x)-L|<\epsilon$ for all $x$ within a distance $\delta$ from $c.$

So if we change $\epsilon$, may be make it smaller, we need to find another $\delta$, most likely smaller than before also, such that $|f\left(x\right)-L|<\epsilon$ inside this new interval around $c$

Figure 3.1:Limit

note: We say $\underset{x\to c}{lim}f(x)$ exist if $\underset{x\to c^{-}}{lim}f(x)=\underset{x\to c^{+}}{lim}f(x)=L$

Example of a function where $\underset{x->0}{lim}f(x)$ does not exist is $f(x)=\{\begin{array}{c}-1x<0\\ 0x=0\\ +1x>0\end{array}$

Figure 3.2:example

note: A function can be defined at a point, but not have a limit at that point (as the example above shows)

3.1.2 Definition of continuous function at a point

A function $f(x)$ is continuous at $x=c$ if it is defined at that point, and if $\underset{x->c}{lim}f(x)$ exist and is equal to $f(c)$

Example: of a function that has $\underset{x->c}{lim}f(x)$ exist, but $f(c)$  is not equal to this limit. hence not continues at $x=c$

Figure 3.3:example

Example of function where $f(c)$ equal the limit at $x=c$, and $\underset{x->c}{lim}f(x)$ exist, hence continues

Figure 3.4:example

3.1.3 Definition of derivative of function $f(x)$ at $c$

if $f(x)$ is continues at $x=c$, then $$f^{\prime }(c)=\underset{x\to c}{lim}\frac{f(x)-f(c)}{x-c}$$

note: The above $f^{\prime }(c)$ is defined only if the limit exist and is the same as we approach $c$ from either side.

Conversely, we say that a function $f(x)$ is differentiable at $x=c$ iff $f^{\prime }(c)$ exist and $f(c)$ is continues.

In other words, $f(x)$ is differentiable at $x=c$ iff $\underset{x\to c^{-}}{lim}\frac{f(x)-f(c)}{x-c}=\underset{x\to c^{+}}{lim}\frac{f(x)-f(c)}{x-c}=f^{\prime }(c)$

note: It is possible for a function to be continues at $c$ but not be differentiable there if the above limit is not the same as we approach $c$ from either side.

Example, $f(x)=\left|x\right|$

Figure 3.5:example

Figure 3.6:example

3.1.4 Intermediate value theorem

on interval $[a,b]$, a continues function assumes all values between $f\left(a\right)$ and $f(b)$