Return to Answer

Update: it answers the different question.

If $1/(N-1)>x> 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps (values at points of jumps is difficult to define, since $p_N$ for $x=1/N$ is $1^{\infty}$).

How to differentiate a tower? Simply fix all but one appearings of $x$, differentiate with respect to this $x$, and then sum up.

Update: it answers a different question.

If $1/(N-1)>x> 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps (values at points of jumps is difficult to define, since $p_N$ for $x=1/N$ is $1^{\infty}$).

How to differentiate a tower? Simply fix all but one appearings of $x$, differentiate with respect to this $x$, and then sum up.

If $1/(N-1)>x> 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps (values at points of jumps is difficult to define, since $p_N$ for $x=1/N$ is $1^{\infty}$).

How to differentiate a tower? Simply fix all but one appearings of $x$, differentiate with respect to this $x$, and then sum up.

Update: it answers the different question.

If $1/(N-1)>x> 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps (values at points of jumps is difficult to define, since $p_N$ for $x=1/N$ is $1^{\infty}$).

How to differentiate a tower? Simply fix all but one appearings of $x$, differentiate with respect to this $x$, and then sum up.

If $1/(N-1)>x\geq 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps.

How to differentiate a tower? Simply fix all but one appearings of $x$, differentiate with respect to this $x$, and then sum up.

If $1/(N-1)>x> 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps (values at points of jumps is difficult to define, since $p_N$ for $x=1/N$ is $1^{\infty}$).

How to differentiate a tower? Simply fix all but one appearings of $x$, differentiate with respect to this $x$, and then sum up.