Fall 2004 #4

Problem Statement: Let f be a continuous function on [a,b] such that f is differentiable at all points of (a,b) except possibly at a single point x_0\in(a,b). If \lim\limits_{x\rightarrow x_0}f'(x) exists show that f'(x_0) exists and that f'(x_0)=\lim\limits_{x\rightarrow x_0}f'(x).

Proof: f'(x)=\lim\limits_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h} by definition. Consider: \lim\limits_{x\rightarrow x_0}f'(x). Applying the definition stated above we get the following:

\lim\limits_{x\rightarrow x_0}f'(x)=\lim\limits_{x\rightarrow x_0}\lim\limits_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}

=\lim\limits_{h\rightarrow 0}\lim\limits_{x\rightarrow x_0}\dfrac{f(x+h)-f(x)}{h}

=\lim\limits_{h\rightarrow 0}\dfrac{f(x_0+h)-f(x_0)}{h}



Reflection: The only part of this proof that I’m worried about is the switching the limits. I’m 90% sure this is allowed since we’re given that f is continuous.

This entry was posted in Analysis, Differentiable. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s