Short and simple true/false tasks from Differentiability, Continuity, and such

Question

These questions come from exams from the previous years. It's not a homework or anything, just preparing for a soon-to-come test.

It's a TRUE/FALSE task with few sentences. Some of them I know answers to, some of them not.

Would anyone help me? I don't really need proofs or very detailed information, just if somebody could try to solve it and double-check my answers :) Thanks!

1. Every continuous function is differentiable - NO
2. Every differentiable function is continuous - YES
3. Every function continuous at $<a,b>$ is integrable at $<a,b>$ - ??
4. Every increasing sequence bounded below is convergent - ??
5. If $x_{0}$ is isolated point of $D_{f}$, then $f$ is not continuous at $x_{0}$ - ??
6. Darboux property can be applied in discontinuous functions - ??
7. $ \forall x \in (a,b)U(c,d) \quad \quad f'(x) > 0 \Rightarrow f$ is increasing at $(a,b)U(c,d)$ - YES
8. Every convergent sequence is bounded - ??
9. Every bounded sequence is convergent - ??
10. Every sequence is continuous function. - YES
11. Every integrable function at $(a, b)$ is also bounded at $(a, b)$
12. Graph of continuous function can have vertical asymptote - YES
13. Domain of $f'$ is contained within domain of $f$ - YES

Answer 1

3. if < a, b> == the interval [a,b] so yes.
4. no. see $a_n = n$
5. YES
6. NO. see the sequance 1, -1
7. NO