Question

I am trying to understand the proof for the theorum that if f is differentiable at x0 then f is continuous at x0. In the class notes it shows limit as x goes to x0 of ((f(x)-f(x0)(x-x0))/(x-x0)) is the same as the limit of ((f(x)-f(x0))/(x-x0)) times (x-x0). Can you factor terms out of limits?

Answer 1

The definition of continuity is \[\lim_{x \rightarrow x _{0}} f(x) = f(x _{0})\]
We rewrite it to read:
\[ \lim_{x \rightarrow x _{0}} f(x) - f(x _{0}) = 0\]
Then you multiply and divide by
\[(x-x _{0})\]
to get
\[\lim_{x \rightarrow x _{0}} (x-x _{0}) * (f(x) - f(x _{0}) / (x-x _{0}) ) \]

Since the 2nd term in the parentheses is the definition of the derivative, we assume that the function is differentiable. And since the first term approaches 0 at the limit, the whole equation approaches zero. Given this, we conclude that if a function is differentiable it is also continuous.

At least that's how I understand it. :)

Answer 2

It sounds reasonable. My only confusion is with the last line of the equation, where
\[\lim_{x \rightarrow x _{0}}(x-x _{0})*(f(x)-f(x _{0})/(x-x _{0}))\] is treated as being \[=\lim_{x \rightarrow x _{0}}(x-x _{0}) * \lim_{x \rightarrow x _{0}}(f(x)-f(x _{0}))/(x-x _{0})\]\[=0*f'(x)\]I did not think f'(ab) was the same as f'(a) * f'(b).

Answer 3

Nevermind - I was confused between limit and derivative for a moment there. Thanks for the help.