Price elasticity of demand always increases with price? Is there a closed-form continuous demand function whose price elasticity of demand decreases with the price?
There are many functions where absolute value of elasticity is decreasing and which are continuous closed form.
One example of such function would be:
$$Q= a-\ln(p), p\geq1 \implies EL = - \frac{1}{p}\frac{p}{a-\ln p} = -\frac{1}{a - \ln p}, $$
which is continuous closed form  and decreasing in price in its absolute value.
Original Answer:
Originally I interpreted the request too mathematically I am leaving it here since other answer references it.
Yes for example consider the following demand function:
$$Q = e^{-p}$$
price elasticity of demand is given by:
$$EL = \frac{dQ}{dp}\frac{p}{Q} = -e^{-p} \frac{p}{e^{-p}} = -p$$
So the demand function has elasticity that is always decreasing in price $p$. The function has closed form, and it is continuous.