High School Continuous but Not Differentiable

[logan3]

Suppose a certain function in continuous at c and (c. f(c)) exists, then which of the two could be false: \displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)}, and \displaystyle f'(c)?

I feel like both could be false, because if the formal derivative at a point exists, then the left and right hand limits much be equal -- but the function could be f(x) = |x|, which means that \displaystyle \lim_{x \rightarrow 0^-} {f(x)} \neq \lim_{x \rightarrow 0^+} {f(x)} (but \displaystyle \lim_{x \rightarrow 0} {f(x)} = 0) and f'(0) = \frac {d}{dx}|0| does not exist. I feel like I'm missing something, cause the nuances around continuity and differentiability have always been confusing (and vague) to me.

Thank-you

 

[Samy_A]

Suppose a certain function in continuous at c and (c. f(c)) exists, then which of the two could be false: \displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)}, and \displaystyle f'(c)?

I feel like both could be false, because if the formal derivative at a point exists, then the left and right hand limits much be equal -- but the function could be f(x) = |x|, which means that \displaystyle \lim_{x \rightarrow 0^-} {f(x)} \neq \lim_{x \rightarrow 0^+} {f(x)} (but \displaystyle \lim_{x \rightarrow 0} {f(x)} = 0) and f'(0) = \frac {d}{dx}|0| does not exist. I feel like I'm missing something, cause the nuances around continuity and differentiability have always been confusing (and vague) to me.

Thank-you

If f is continuous in c, then $\displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)} =f(c)$.

For $f(x)=|x|$, what you have is that $\displaystyle \lim_{x \rightarrow 0^-} \frac{f(x)}{x}=\lim_{x \rightarrow 0^-} \frac{-x}{x}=-1$ and $\displaystyle \lim_{x \rightarrow 0^+} \frac{f(x)}{x}=\lim_{x \rightarrow 0^+} \frac{x}{x}=1$. That's why the function has no derivative in 0.

 

[HallsofIvy]

Suppose a certain function in continuous at c and (c. f(c)) exists, then which of the two could be false: \displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)}, and \displaystyle f'(c)?

I feel like both could be false, because if the formal derivative at a point exists, then the left and right hand limits much be equal -- but the function could be f(x) = |x|, which means that \displaystyle \lim_{x \rightarrow 0^-} {f(x)} \neq \lim_{x \rightarrow 0^+} {f(x)}

No, this is wrong. Both \displaystyle \lim_{x \rightarrow 0^-} {f(x)} and \lim_{x \rightarrow 0^+} {f(x)} exist and are equal to 0.
If \lim_{x\to a} f(x) exists then it must be true that \lim_{x\to a^-} f(x) and \lim_{x\to a+} f(x) exist and are equal. Perhaps you are thinking of the fact that the limits of the derivatives are not equal: \displaystyle \lim_{x \rightarrow 0^-} {f'(x)} \neq \lim_{x \rightarrow 0^+} {f'(x)}.
(Note f'(x), not f(x).)

(but \displaystyle \lim_{x \rightarrow 0} {f(x)} = 0) and f'(0) = \frac {d}{dx}|0| does not exist. I feel like I'm missing something, cause the nuances around continuity and differentiability have always been confusing (and vague) to me.

Thank-you