Every Differentiable Function is Continuous, Proof

Everyone knows this, but I bet a lot of people can't prove it! I'm also sure that plenty of people can, but just in case you have never seen the proof, here it is anyways!

Ok so suppose f is differentiable at c.

This means that  lim x->  (f(x) - f(c))/(x-c) exists and the derivative of f at c is this limit;i.e.,

f'(c) = lim x->  (f(x) - f(c))/(x-c).

Ok so now we need to show f is continuous at c, so we need to show lim x->c f(x) = f(c).

Instead we will show that lim x->c (f(x) - f(c)) = 0 and then it will follow from there.

Well the trick is to just rewrite the above limit and somehow use what we already have.

lim x->c (f(x) - f(c))=
lim x->c (f(x) -f(c))/(x-c) * (x-c) =
(lim x->c (f(x) - f(c))/(x-c)) * lim x->c (x-c) = f'(c)*0 = 0.

Hence we showed that,

lim x->c (f(x) - f(c)) = 0.

Then,

lim x->c f(x)
lim x->c (f(x) - f(c) + f(c))  <--- just adding 0 which is the same as -f(c) + f(c)
lim x->c (f(x) - f(c)) + lim x->c f(c) <--- since both limits exist we can do this
0 + f(c)
f(c)

Done!