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Testing a resistor
Usually when a resistor fails they either increase in value or not open up at all. 
You can check the resistance of a resistor with an ohmmeter. If the resistor is in circuit, you will generally have to isolate the resistor so you are measuring only the resistor, not other components in the circuit.
Using analog meter to check resistor on board often produces an incorrect reading. This is due to the reason that the output from the analog meter is from 3 volt to 12 volt. The voltages are quite high and it can trigger the semiconductor devices around the resistors such as diode, transistor and ICs. Semiconductors only need voltage of 0.6v in order to conduct. Since the output voltage from the analog meter is higher than the semiconductors, it won't give you an accurate reading. In order to measure resistors while it still in circuit, you need to get a digital multimeter that have the output of less than 0.6v. This is to avoid conducting the semiconductor devices around the circuit that you want to check. 

If you connect your digital meter leads across a resistor in a circuit and it measures higher than it should, then you know the resistor is either open or has gone up in value. Other circuit components cannot possibly increase the value of a resistor any parallel circuit could only make the resistance reading lower.  In rare cases, sometimes an undischarge capacitor can cause the measurement higher than it should be. 
How to discharge a capacitor
Typical values of this of filter capacitors are
220uf 250volt, 150uf 400volt and 330uf 450volt

 If you are going to work on the power supply circuits, it's a good idea to discharge that capacitor first.

Method #1
Discharge it with a screw driver (not recommended).
The reason for not using the screw driver to discharge a capacitor is because the printed circuit board or circuitry can be damage due to the spark generated while discharging the high voltage capacitor.

Method #2
Place the leads of a socketed 100 watt electric bulb on the leads of the capacitor. The light bulb will act as an indicator to see if the capacitor still holds the charge.

Method #3
Place the leads of a high wattage resistor on the leads of the capacitor.
I use a 2.2k ohm 10 watt resistor to discharge the high voltage capacitor in a switch mode power supplies. 
Testing Transformers
There is two types of transformers in the market
Linear Transformers 
Switch Mode Power Transformers 

Linear transformers are mostly used in radio, project kits, VCD and etc Switch mode transformers can be found in Monitor, Tv, Fax etc.
A transformer consists of two or more coils wound on the same core. For power transformers, the core material is usually iron. For a radio-frequency, the core material is usually ferrite or air. The basic property of a transformer is to change AC voltage. A transformer cannot change direct current voltage. A step down transformer has a lower AC output voltage at its secondary winding than the AC input voltage to its primary winding. A step up transformer has a higher secondary than primary voltage. The transformer laminations or coating of shellac, enamel or varnish is to insulate adjacent turns from shorts between winding.  If too much current goes through a coil or winding, the winding heats up and can either open up completely, or the insulation between turns of wire can break down, causing the coil to be partly or completely shorted. You can check continuity of any primary or secondary transformer winding.

Linear and Power Testing
Linear Apply the AC voltage to the primary winding and expect output voltage at the secondary side. No or low output voltage means the transformer winding has open or shorted winding. Look for any evidence of overheating, such as darkened or blackened areas or a burnt smell. 
Power You cannot apply ac voltage to the equipment and expect the desire output at the secondary side without connect to load otherwise over voltage will happen (because you have removed all the secondary load) and will blow the power section which consist of power fet, pwm ic and etc.

The best way to test a switch mode power transformer is first remove it from the board. Trace from the big filter cap the positive pin to see where the circuit goes. It will go to one of the pin of the transformer. Then look for the power middle pin which is the Drain and it will lead to another pin of the transformer. Checking the resistance of the primary winding only can reveal whether it is open but it can’t check if it has developed a shorted winding. Only by using a ringer test you will know if it shorted or not. Connect the probes to the primary winding and the result are instantly seen 4-8 bar represent a good winding, 1-2 bar represent partial short and if the led goes off means there is a heavy short in the winding. It can be use to check B+ COIL, horizontal yoke coil and flyback primary winding and ballast (choke). 
Testing a Diode
A rectifier diode can fail in one of the four ways.
(1) Open (2) Shorted (3) Leaky (4) Breakdown in full operating voltage

An analog multimeter or digital multimeter can be used to check for all the first three conditions except Failute #4 where the breakdown is in full operating voltage.  You can't always be certain if a diode is good or bad if you perform in-circuit test, because of back circuits through other components.

Set the analog meter to x1 ohms to check for current diode leakage reverse and forward testing. Connecting the black probe of your meter to the cathode and red probe to the anode, the diode is reverse biased and should look like an open reading. Connecting the red probe of your meter to the cathode and black probe to the anode, the diode is forward biased and the meter should read some value of resistance. 

If you have two readings then most probably the diode is shorted or leaky. If you don't get any reading either forward or reverse bias, the diode is considered open.  The real problem when checking a diode using the diode test function of a digital meter is that an open or leaky diode, the meter sometimes reads ok (0.6). This is due to digital meter diode test output voltage is around 500mv to 2v. An analog meter set to x1 ohms have output about 3V(the two 1.5V battery you installed in the meter). The 3V voltage is enough to show you the accurate reading of a diode when under test.  You now have to set your meter to x10K to test the diode again. The output voltage ofx10k ohms is about 12V(the 9v battery in your meter-1.5v+1.5v +9v=12v). The diode under test should show only one reading. 

If the meter showed one reading then the diode under test is good. If it has two readings then most probably the diode is either shorted or leaky. The digital meter can't test it because the output from the meter is only500mv to 2V. If a diode breakdown when under full operating voltage, there is no way to testing diode.

You may also use a light bulb to check a Transformer. Remove one of the
primary winding pin and connect a 60 watt light bulb. One end to the
circuit while the other to the transformer pin. If the bulb light and didn't
goes off suspect a shorted winding and if the light goes off, the winding
is ok. 
Electric Diode
Typical Electrolyte
Capacitor