Continuity & One-Sided Limits

For $f$ to be continuous at $x=c$, which statement must be true?

1. Only $f(c)$ exists
2. Only the limit exists
3. $f(c)$ exists, the limit exists, and $\lim_{x\to c}f(x)=f(c)$
4. The graph must be a line

What value repairs the missing point at $x=1$?

1. $0$
2. $1$
3. $2$
4. No value can repair it

Answer: $2$

The nearby rule is $x+1$, whose limit at $1$ is $2$.

If $\lim_{x\to a^-}f(x)$ and $\lim_{x\to a^+}f(x)$ are finite but unequal, what type of discontinuity occurs at $x=a$?

1. Removable
2. Jump
3. Continuous
4. A hole only

Where can $f(x)=\dfrac{x+1}{x^2-9}$ fail to be continuous?

1. Only at $x=0$
2. $x=\pm3$
3. Everywhere
4. Nowhere

Which continuity equation determines $m$?

1. $6+m=8$
2. $m=8$
3. $6=m$
4. $6+m=9$

Answer: $6+m=8$

The left limit is $6+m$ and the right limit and value are $8$.

At a breaking point $x=c$ of a piecewise function, what should you compare?

1. Only $f(c)$
2. Left limit, right limit, and $f(c)$
3. Only the slopes
4. Only the domain

At $x=3$, what is the common limit and value of $|2x-6|$?

1. $3$
2. $6$
3. DNE
4. $0$

Answer: $0$

Both pieces approach $0$, which equals the function value.

Is $x\lfloor x\rfloor$ continuous at $0$?

1. No, because floor always forces a jump
2. Yes: both limits and the value are $0$
3. Only from the left
4. Only from the right

Answer: Yes: both limits and the value are $0$

Both one-sided limits are $0$, and the value at zero is $0$.

For $f(x)=\lfloor x\rfloor$, compare the one-sided limits at an integer $n$.

1. Both equal $n$
2. $\lim_{x\to n^-}f(x)=n-1$ and $\lim_{x\to n^+}f(x)=n$
3. Both equal $n-1$
4. Both do not exist

To be continuous on $[a,b]$, what is required at $a$?

1. $\lim_{x\to a^-}f(x)=f(a)$
2. $\lim_{x\to a^+}f(x)=f(a)$
3. $f(a)=0$
4. No endpoint condition

What does IVT alone prove for this polynomial on $[0,1]$?

1. Exactly one zero
2. The zero equals $1/2$
3. At least one zero in $(0,1)$
4. No zero

Answer: At least one zero in $(0,1)$

The polynomial is continuous and its endpoint values straddle zero, so at least one zero lies inside.

If $f$ is continuous on $[a,b]$ and $K$ lies between $f(a)$ and $f(b)$, IVT guarantees:

1. Exactly one $c$
2. $f(c)=K$ for at least one $c\in[a,b]$
3. $f(c)=0$ always
4. $f$ is linear

After the first midpoint test, which interval retains the sign change?

1. $[1,1.5]$
2. $[1.5,2]$
3. $[1,1.25]$
4. $[0,1]$

Answer: $[1,1.5]$

The signs at $1$ and $1.5$ are opposite, so this is the next bracket.

For a continuous function with opposite endpoint signs, the midpoint output is nonzero. Which half-interval should be kept?

1. Always the left half
2. Always the right half
3. The half whose endpoint values have opposite signs
4. The half with the larger function values