With capacitors, they don't show continuity. Your best bet is to check the continuity between the pad and the track you're soldering it to, and that the solder joint is shiny. Also, if they're the little can type ones, make sure you get them the right way around as they're polarised (usually marked with either + and -, or the minus side is marked with a black strip).
With LED's, they're diodes, which are also polarised. Continuity checks on a multimeter may not light an LED up (may not have enough voltage). If others do light up but not this one, then you may have it backwards (easy with such a tiny component). You can test that by simply reversing your test leads.
FWIW: A simple test for the polarity of a LED is 2 x AA or AAA batteries in a small battery holder, and a small resistor of about 180 ohms. Connect the red lead on the battery holder to the resistor. Then connect the LED between the other end of the resistor and the black lead of the battery holder. If it doesn't light up, reverse the pins on the LED. Once it lights up, the pin on the LED connected to the resistor is the Anode (triangle side of the schematic diagram) and the black lead is connected to the Cathode (the bar side of the schematic diagram, and usually marked on the PCB as a bar).
With LED's, they're diodes, which are also polarised. Continuity checks on a multimeter may not light an LED up (may not have enough voltage). If others do light up but not this one, then you may have it backwards (easy with such a tiny component). You can test that by simply reversing your test leads.
FWIW: A simple test for the polarity of a LED is 2 x AA or AAA batteries in a small battery holder, and a small resistor of about 180 ohms. Connect the red lead on the battery holder to the resistor. Then connect the LED between the other end of the resistor and the black lead of the battery holder. If it doesn't light up, reverse the pins on the LED. Once it lights up, the pin on the LED connected to the resistor is the Anode (triangle side of the schematic diagram) and the black lead is connected to the Cathode (the bar side of the schematic diagram, and usually marked on the PCB as a bar).