(i) and (iv) are true

(i) and (ii) are true

(i), (ii), and (iv) are true

only (iv) is true

(i), (ii), and (iii) are true

All statements are true

None of the statements are true

(ii) and (iv) are true

f(z) is nowhere differentiable. 

f(z) is nowhere analytic.

None of the others.

f(z) is continuous everywhere on \mathbb{C}.

There exists z_0\in\mathbb{C} such that f(z) is differentiable at z_0.

There exists z_0\in\mathbb{C} such that f(z) is analytic at z_0 .

(i), (iii) and (iv) are correct

Only (iii) is correct

(i) and (ii) are correct

Only (i) is correct

Only (ii) is correct

(i) and (iii) are correct

Only (iv) is correct

All of the statements are correct

f(z) is nowhere analytic.

There exists z_0\in\mathbb{C} such that f(z) is continuous at z_0.

None of the others.

There exists z_0\in\mathbb{C} such that f(z) is differentiable at z_0.

f(z) is discontinuous at 0.

There exists z_0\in\mathbb{C} such that f(z) is analytic at z_0.

(iii) and (iv) are true

Only (v) is true

(i), (iii), (iv), and (v) are true

(iii), (iv), and (v) are true

All of the statements are true

None of the statements are true

u_n(x) converges to u(x) uniformly for all x\in\mathbb{R}.

u'_n(x) converges to 1 for x>0 and to -1 for x < 0, and the convergence is uniform.

u_n(x) is differentiable for all x\in\mathbb{R}.

u(x) is differentiable for all x\in \mathbb{R}.

\lim_{n\to\infty}\int_{-1}^1u_n(t)dt=\int_{-1}^1 u(t)dt.

\int_0^x u_n(t)dt is uniformly converging to \int_0^x u(t)dt for all x\in [0,b]\subset \mathbb{R}.

\lim_{n\to\infty}\int_0^1u_n(t)dt=\int_0^1 u(t)dt.

\liminf_{n\to\infty} u_n=\limsup_{n\to\infty} u_n=L if and only if \lim_{n\to\infty} u_n=L.

For u_n=\sin n we have \liminf_{n\to\infty} u_n=-1 and \limsup_{n\to\infty} u_n=1

\liminf_{n\to\infty} u_n\leq \limsup_{n\to\infty} u_n .

For u_n=2+(-1)^n\left(1+\frac{1}{n}\right ) we have \liminf_{n\to\infty} u_n=1 and \limsup_{n\to\infty} u_n=3

\limsup_{n\to\infty} u_n always exists.

\liminf_{n\to\infty} u_n always exists.

For u_n=(-1)^n we have \liminf_{n\to\infty} u_n=-1 and \limsup_{n\to\infty} u_n=1

None of the other responses.

\dfrac{d}{dz}\left(g\left(f(z)\right)\right)=6z^6

For each z\in \mathbb{C}, f'(z)\ne g'(z).

g\left(f(2i) \right)=63

\dfrac{d}{dz}\left(f(z)g(z)\right)=3z^2\left((z-i)^2 + 1\right) - 2(z^3 - i)(z + i)

f'(2i)+g'(3)=-6+4i

If f(z_0) or \ lim_{z\to z_0}f(z) does not exist then f is discontinuous at z_0.

If \lim_{z\to z_0}f(z)=2025 then f is discontinuous at z_0.

If f(z_0) does not exist then f is discontinuous at z_0.

If \lim_{z\to z_0}f(z) does not exist then f is discontinuous at z_0.

If \lim_{z\to z_0}f(z)=2025 then f can be continuous at z_0.

If f(z) is discontinous at z_0 then \lim_{z\to z_0}f(z) does not exist.

If f(z) is discontinuous at z_0 then f(z_0) does not exist.

\frac{1}{R}=\lim_{n\to \infty} \left | \frac{a_n}{a_{n+1}}\right | when the limits exists.

If a_n=\frac{1}{n!} then R=0.

Taylor series for f and Taylor series for \int_a^x f(t)dt have the same R.

Series is diverging for |x-a|>R.

\frac{1}{R}=\lim_{n\to \infty} |a_n|^{\frac{1}{n}} when the limits exists.

Series is absolutely converging for |x-a| < R.

Series is uniformly converging for |x-a|\leq \rho where \rho < R .

If a_n=n! then R=\infty.

Taylor series for f and Taylor series for f' have the same R.

Series may or may not converge for |x-a|=R.