a function is discontinuous but its limit is differentiable

[david]

is there exist such a sequence {fn} such that is for each fn is discontinuous but its limit function is differentiable?

 

[HallsofIvy]

The wording is very strange. What do you mean by the 'limit' of a single function?

I suspect you mean that \(\displaystyle \{f_n(x)\}\) is a sequence of discontinuous functions and want to know if \(\displaystyle \lim_{n\to\infty} f_n(x)\) can be continuous. Think about \(\displaystyle f_n(x)= 0 \text{ if } x< n\), \(\displaystyle f_n(x)= 1 \text{ if } x\ge n\). What is the limit of that sequece?

 

[david]

can lim_{n-> inf} f_n​(x) be differentiable

 

[daon2]

Did you attempt the above example? Here's another:

\(\displaystyle f_n(x) =
\begin{cases}
x^2, & \text{if } x\neq 0 \\
\dfrac{1}{n}, & \text{if } x=0
\end{cases}\)

 

[DrPhil]

can lim_{n-> inf} f_n​(x) be differentiable

Consider whether or not you can evaluate

\(\displaystyle \displaystyle \lim_{h\to 0} \left[ \dfrac{\lim_{n\to \infty} f_n(x+h) - \lim_{n\to \infty} f_n(x)}{h} \right] \)