Use the $\epsilon$-$\delta$ definition of a limit to prove that:

Solution

(a) $f(x)=A$ is the constant function whose graph is a horizontal line. Given any $\epsilon >0$, we must find $\delta >0$ so that whenever $0 \lt \vert x-c\vert \lt \delta ,$ then $\left\vert f(x)-A\right\vert \lt \epsilon$.

Since $\left\vert A-A\right\vert =0,$ then $\left\vert f(x)-A\right\vert \lt \epsilon$ no matter what positive number $\delta$ is used. That is, any choice of $\delta$ guarantees that whenever $0 \lt \vert x-c\vert \lt \delta$, then $\left\vert f(x)-A\right\vert \lt \epsilon$.

(b) $f( x) =x$ is the identity function. Given any $\epsilon >0$, we must find $\delta$ so that whenever $0 \lt \vert x-c\vert \lt \delta$, then $\vert f(x)-c \vert =\vert x-c\vert \lt \epsilon$. The easiest choice is to make $\delta =\epsilon$. That is, whenever $0 \lt \vert x-c\vert \lt \delta =\epsilon$, then $\vert f(x)-c \vert \hspace{-6.5pt}\underset{\underset{\color{#0066A7}{\scriptsize\hbox{f(x) = x}}}{\color{#0066A7}{\uparrow}}}{=}\hspace{-6.5pt} \vert x-c\vert \lt \epsilon$.