Specific heat of a substance is defined as the amount of heat required to increase the temperature of a unit mass (of this substance) by 1 degree Celsius. Correspondingly, it would be the amount of heat released when the temperature of a unit mass of a substance drops by 1 degree Celsius.
If the specific heat of a substance is infinite, then, an infinite amount of heat is required to increase the temperature of a unit mass by 1 degree Celsius. In other words, we can supply all the heat, we want to, to this substance and its temperature will not change (that is, increase). Thus, the statement that "the temperature will not change whether the heat is given or taken" is accurate. There is no evidence that such a material will produce or consume heat, although if that is happening,we will be able to detect any temperature change in this substance.
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