When there is a significant change in temperature, the frog can jump out quickly. And when the change in water temperature is very small, it is difficult for the frog to notice and lose the chance to escape. "The rationale behind the "boiling frog in warm water" story makes sense, except that the warming conditions that can kill a frog aren't quite the same as what we generally understand as "boiling.
"Boiling a frog in warm water" comes from a famous experiment done by scientists at Cornell University in the U.S.: researchers threw a frog into boiling water. In the nick of time, the frog jumped out of the pot and escaped unharmed; afterward, the researchers put the frog into a pot full of cold water and then slowly warmed it up.
At first, the frog was very proud of itself, relaxing in the warm, cozy water, until it felt the water was too hot to bear, and then tried to jump out of the water but had no strength in its limbs, and could not jump, and eventually the frog was boiled to death in the hot water.
The purpose of these "boiling" experiments was to examine the relationship between temperature and neural reflexes. The reason the frogs jumped out at faster heating rates is well understood: the thermal stimulus caused the frogs to jump away as a stress response.
At slower rates of heating, for reasons that resemble "sensory adaptation," the frog adapts to the stimulus by sustaining small changes in temperature, and its reflexes become less stressful until it reaches a threshold temperature at which it can tolerate the heat and is unable to escape.
Conclusion: In the case of a significant temperature change, the frogs could jump out quickly. When the change in water temperature is very small, the frogs are hard to notice and lose the chance to escape. The rationale behind the story of the "boiling frog in warm water" has some truth to it.
In March 2014, Zheng Yu and four other students from Guangzhou University's School of Life Sciences conducted this experiment. They put a certain amount of water at 20 degrees Celsius in a beaker and placed the tiger frog in it to heat it up, and when the temperature of the water in the beaker increased, they found that the frog would jump out of the water, instead of being boiled to death in the water all the time as we know it.
Multiple experiments showed that when the beaker contained 440 milliliters of water, the average temperature of the water when the frog jumped out was 30 ℃, while when the beaker contained 800 milliliters of water, the average temperature of the water when the frog jumped out was 32.8 ℃. The temperatures that the frogs could tolerate in the water varied slightly with the amount of water, but they all escaped as the temperature increased.