05 | September | 2008 | Continuous Everywhere but Differentiable Nowhere

I asked a question to the two AP calculus teachers today, and I think we’ve concluded that we each aren’t 100% sure of the answer. It’s one of those questions that seems so basic that how could we not be sure?

I’m going to put up two graphs (of the square root of x). Can you tell me what intervals the following function are increasing on? [Notice the difference: Function A is defined at x=0, Function B is not defined at x=0.]

FUNCTION A

FUNCTION B

According to Anton’s Calculus text, it says:

Let $f$ be a function that is continuous on a closed interval $\text{[}a,b\text{]}$ and differentiable on the open interval $(a,b)$ .

If $f'(x)>0$ for every value of $x$ in $(a,b)$ , then $f$ is increasing on $\text{[}a,b\text{]}$ .
If $f'(x)<0$ for every value of $x$ in $(a,b)$ , then $f$ is decreasing on $\text{[}a,b\text{]}$ .

According to Rogawski’s Calculus text, it says:

Let $f$ be a differentiable function on the open interval $(a,b)$ .

If $f'(x)>0$ for $x \in (a,b)$ , then $f$ is increasing on $(a,b)$
If $f'(x)<0$ for $x \in (a,b)$ , then $f$ is decreasing on $(a,b)$

So my questions are: Are these two different definitions? I’m not teaching AP Calculus, but would this even be an issue for the AP exam? And why am I so not getting this?

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now

Day: September 5, 2008

Mid-Day Calculus Question