Have scientist invented a capacitor that charges instantanously but
discharges slowly like a battery, like  many days or weeks. And what
is this capacitor called.

The charge on a capacitor is related to its capacitance (C), time (t)
and the resistance in the circuit (R). The relationship is given by
Q=Q'e^-(t/CR)  where Q' is the initial charge and Q is charge at any
time. CR is known as the time constant  of the circuit ad is the time
taken for the chrge on the capacitor to fall to 1/e of its initial
value.
From this equation you can see that any capacitor follows an
exponentional charging and discharging curve. Hence discharge rate can
be controlled by varying the circuit resistance. High resistance means
slow discharge.

Regarding your comment about instantaneous charging. As you probably
know nothing happens instantaneously. Even light has a finite speed as
do electrons. If you have a low circuit resistance you can charge
quickly but never instantaneously.
Nick

I would build a circuit to accomplish this. Use a diode and small or
no resistor in series to charge the cap very quickly, then use a
larger resistor to limit the current flow during discharge.

Hi,

With a capacitor being used as a battery there is three special issues:

- Serial resistance: this the resistance that the capacitor presents
to the circuit that limits the maximum current that you are able to
get from (or put into) the capacitor, and gives looses and produce
heat.
- Self discharge: All the capacitor (as the batteries) have a self
discharge rate that can be significative depending on manufacturing
technology.
- Energy density: The capacitor does not use to have high energy
density storage values. To make calculus, the energy stored in a
capacitor is related to the voltage and the capacity:
     Energy = C * V^2 / 2

The best suited capacitor type to be used as a battery is the one
named as "supercap". Marketing gives them several different names,
like ultracap, dynacap, Dual layer capacitors,...

The supercap has the biggest of all types energy density, close to
some battery standards.
It also has low self discharge rate.
There may be some problem with serial resistance depending on your
demands. But now there is models with very low serial resistance. 
Depends on the voltage and capacity your application needs.

It is perfect to be, for example, a backup source to semiconductor
memories or clocks.
It is being studied as a battery source to electrical vehicles.
The charging rate use to be far faster than batteries.

Hope this helps,

mlopez.