Continuity at a Boundary

Q.

What are the $3$ conditions of continuity?

Q.

What are differentiability and continuity?

Q.

The function f(x) is continuous in the interval (a, b) then which among the following is true?

1. Function may or may not have a value at x=a, b

2. Function should exist at x=a but may not be continuous

3. f(a) = $\underset{x\to a^{+}}{lim}f\left(x\right)$

4. f(a) > f(b)

Q. Let $f:R\to R$ be a function defined as
$f\left(x\right)=\{\begin{array}{cc}5,& \text{if}x\le 1\\ a+bx,& \text{if}1<x<3\\ b+5x,& \text{if}3\le x<5\\ 30,& \text{if}x\ge 5\end{array}$
Then, which of the following is correct regarding the function $f$ :

1. continuous if $a=-5$ and $b=10$
2. continuous if $a=5$ and $b=5$
3. continuous if $a=0$ and $b=5$
4. not continuous for any values of $a$ and $b$

Q.

Function f(x) is defined in [a, b]. If the function is continuous throughout the interval [a, b] then which among the following are correct

1. $\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$

2. $\underset{x\to a}{lim}f\left(x\right)=\underset{x\to b}{lim}f\left(x\right)$

3. $\underset{x\to b}{lim}f\left(x\right)=f\left(a\right)$

4. $\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{x\to a^{-}}{lim}f\left(x\right)$

Q. Let $a,b\in R,(a\ne 0).$ If the function $f$ defined as
$f\left(x\right)=\{\begin{array}{c}\frac{2x^2}{a},0\le x<1\\ \\ a,1\le x<\sqrt{2}\\ \\ \frac{2b^2-4b}{x^3},\sqrt{2}\le x<\infty \end{array}$
is continuous in the interval $[0,\infty ),$, then an ordered pair $(a,b)$ is:

1. $(\sqrt{2},1-\sqrt{3})$
2. $(-\sqrt{2},1+\sqrt{3})$
3. $(-\sqrt{2},1-\sqrt{3})$
4. $(\sqrt{2},-1+\sqrt{3})$

Q.

If a function f(x) is defined in x $ϵ$ [a, b], then f(x) is continuous at a if

1. ${lim}_{x\to a}-f\left(x\right)={lim}_{x\to a^{+}}f\left(x\right)=f\left(a\right)$

2. ${lim}_{x\to a^{+}}f\left(x\right)=f\left(a\right)$

3. Function will be continuous in any case

4. Function will be discontinuous in any case