I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and strictly weaker than the sup-norm topology (the topology of uniform convergence), $\tau_\infty$. Let $\phi:l^\infty \rightarrow l^\infty$ be defined as $$ \phi(x_1,x_2,\dots)=(f(x_1),f(x_2),\dots) $$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous (for the usual topology of $\mathbb{R}$), strictly increasing and $f(0)=0$.
I proved that $\phi:(l^\infty,\tau_p) \rightarrow (l^\infty,\tau_p)$ and $\phi:(l^\infty,\tau_\infty) \rightarrow (l^\infty,\tau_\infty)$ are continuous.
Is it possible to show that $\phi:(l^\infty,\tau) \rightarrow (l^\infty,\tau)$ is continuous?