Expert: Paul Klarreich - 5/8/2006

Question
Hi,

I'm trying to prove that a function f is continuous at a point a given that it is diffentiable at a:

What I've got so far:

a) f is differentiable at a if lim (x->a) (f(x)-f(a))/(x-a) exists at a.

b) f is continuous at a if lim (x->a) f(x) = f(a).

Actually, that's about all I've got! I'm having real trouble linking the two up, even though it looks like it should be a short job...

Any help would be really apprectiated,

Best wishes,
James

Answer
Hi, James,
 
You wrote:
Subject:  Proof of continuity given differntiability at a point.

I'm trying to prove that a function f is continuous at a point a given that it is differentiable at a:

What I've got so far:

a) f is differentiable at a if lim (x->a) (f(x)-f(a))/(x-a) exists at a.

b) f is continuous at a if lim (x->a) f(x) = f(a).

Actually, that's about all I've got! I'm having real trouble linking the two up, even though it looks like it should be a short job...

Any help would be really appreciated,

Best wishes,
James
-------------------------------------------
FIRST: BE SURE TO VIEW THIS IN A FIXED-SIZE FONT.

Try this:

              f(x) - f(a)
f(x) - f(a) =  ------------ (x - a), if x /= a.  [not =]
                 x - a

Now use some properties of limits, such as:

Lim (AB) = lim A  lim B  -- Product property.
Lim (A+B) = lim A + lim B  -- Sum  ...
Lim (A-B) = lim A - lim B  -- Difference ...
Lim c  = c                 -- constant ...

Now, where all the limits are as x->a, we write:

                        f(x) - f(a)
lim (f(x) - f(a)) = lim [----------- (x - a)]
                           x - a

By the product property:
                       f(x) - f(a)
lim (f(x) - f(a)) = lim ----------- lim (x - a)
                          x - a

By the definition of the derivative, and the assumption that f is differentiable at a,

lim (f(x) - f(a)) = f'(a)  lim (a - a)

lim (f(x) - f(a)) = f'(a) * 0

lim (f(x) - f(a)) = 0

By the difference property:
lim f(x) - lim f(a) = 0
lim f(x) = lim f(a)

And, finally, by the constant property:
lim f(x) = f(a)