A pump to be vertically suspended and mounted in a container body for discharging liquid contained in the container body, comprising: a cylinder including: a suction valve at a bottom thereof; and an actuator member urged upward and depressible with respect to the cylinder, wherein the actuator member includes: a stem having a bottomed tubular shape urged upward and depressible; a depression head urged upward and relatively displaceable upward and downward with respect to the stem; and an annular piston provided around an outer circumference of the stem for relative upward and downward displacement with respect to the stem, the depression head has: a valve chamber provided with a discharge opening at a tip thereof; the annular piston has a lower end forming a discharge valve in cooperation with a discharge valve seat protruding from the outer circumference of the stem and is configured to be pressed against the valve seat to close the discharge valve at a lowermost displacement position and displaced off the discharge valve seat to open the discharge valve at an uppermost displacement position, and characterised in that the depression head further comprises an opening-closing valve member disposed in the valve chamber in a manner such that the opening-closing valve member is urged toward the discharge opening, and a lever member connected at an upper end thereof to a rear end of the opening-closing valve member, and is configured such that, when depressed, the lever member of the depression head operates to open the discharge opening by displacing the opening-closing valve member and that a resistive force against the depression of the depression head with respect to the stem is smaller than a resistive force against the depression of the stem itself, and the cylinder further comprises a first resilient member urging the stem; and a second resilient member urging the opening-closing valve member, wherein resilience of the second resilient member is selected to be smaller than resilience of the first resilient member. a cylinder including: a suction valve at a bottom thereof; and an actuator member urged upward and depressible with respect to the cylinder, the actuator member includes: a stem having a bottomed tubular shape urged upward and depressible; a depression head urged upward and relatively displaceable upward and downward with respect to the stem; and an annular piston provided around an outer circumference of the stem for relative upward and downward displacement with respect to the stem, the depression head has: a valve chamber provided with a discharge opening at a tip thereof; the annular piston has a lower end forming a discharge valve in cooperation with a discharge valve seat protruding from the outer circumference of the stem and is configured to be pressed against the valve seat to close the discharge valve at a lowermost displacement position and displaced off the discharge valve seat to open the discharge valve at an uppermost displacement position, and characterised in that the depression head further comprises an opening-closing valve member disposed in the valve chamber in a manner such that the opening-closing valve member is urged toward the discharge opening, and a lever member connected at an upper end thereof to a rear end of the opening-closing valve member, and is configured such that, when depressed, the lever member of the depression head operates to open the discharge opening by displacing the opening-closing valve member and that a resistive force against the depression of the depression head with respect to the stem is smaller than a resistive force against the depression of the stem itself, and the cylinder further comprises a first resilient member urging the stem; and a second resilient member urging the opening-closing valve member, wherein resilience of the second resilient member is selected to be smaller than resilience of the first resilient member.