  # differentiability vs. continuity? HELP please!

<p>what is the difference between proving a function is continuous and proving something is differentiable? are there any good sites that help with these topics?</p>

<p>for continuous make sure the function doesn't have any holes in it (like a 0 in the denominator) and for differentiability make sure the derivative doesn't have any holes in it...if you graph them, it's continuous if you can draw it without lifting your pencil and it's differentiable if it doesnt have any jagged edges. i think thats right.</p>

<p>Well, actually that's not always true. If you have a closed function, its endpoints are continuous, but no differentiable.</p>

<p>for a function to be differentiable over a closed period, it must also be continuous over that period as well.</p>

<p>For a function to be continuous at a point "p" lets say, the limit as x approaches p from the right must equal the limit as x approaches p from the left, which must equal f(p). In other words, f(p) = lim x->p+ = lim x->p- if a function is continuous at point "p."</p>

<p>For a function to be differentiable at point "p", the left and right hand derivatives of the function approaching that point must be equal. Differentiability implies continuity at a point. HOWEVER, notable exceptions of "differentiability implies continuity" include a corner, cusp, horizontal tangent, or discontinuity. These include absolute value functions, x to a fractional power (e.g. x^(2/3) or x^(1/3)), and certain rational functions (e.g. (x^2 - 1)/(x + 1)).</p>