Intuitively, function is continuous at $x=c$ if the graph is connected when you zoom in enough near $x=c.$ A function is differentiable at $x=c$ if when you zoom in$,$ the graph looks like the graph of a linear function $($straight non$-$vertical line$).$

$($a$)$ Sketch the graph of function that is continuous, but not differentiable, at $x=1.$

$($b$)$ If a function is continuous at $x=c,$ does it have to be differentiable at $x=c?$

$($c$)$ If a function is differentiable at $x=c,$ does it have to be continuous at $x=c?$