Also, what is the easiest way to find out whether a function is continuous or differentiable when given a function equation?
From the fundamental definition of the continuity:
A function f is continuous at x=a iff:
for every e>0 there exists a d>0 such that:
whenever |x - a| < d,
then |f(x) - f(a)| < e
So for example if you're asked to prove from the fundamental definition that y=x^2 is continuous at x=0:
whenever |x - 0| = |x| < d,
we require |x^2 - 0^2| = x^2 < e
since |x| >= x^2 for x near 0,
it will suffice to require that |x| < e, in which case we can choose:
d = e
therefore y=x^2 is continuous at x=0.
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Obviously this isn't the easiest way to work out if a function is continuous, you only use this method if you're asked to prove a function is continuous from the fundamental definition. Otherwise see if it is composed of functions you know that are continuous.
To prove a function f is differentiable at x=a, it is quite easier.
f is differentiable at x=a if:
lim (h->0) {f(a+h) - f(a)}/ h exists