Sampling distribution of a statistic is the probability distribution of possible values of the statistic for repeated samples of the same size taken from the same population.
How this is useful is that we can use a sample statistic and make inferences about what the value of the corresponding population parameter is.
A function is continuous at a point χ when no matter how small we make the interval around f(χ) we can make an interval around χ where all values of x in the interval mean f(x) fall within the required interval. This means we can achieve whatever precision we need around f(χ).
To show continuity, relate the size of the interval around f(χ) to the necessary size of the interval around χ. It needs to be related because if we change the required interval around f(χ) we need to change the interval around χ accordingly.
A function is not continuous at χ when for some interval around f(χ) we can’t define an interval around χ so all values within that interval mean f(x) is within the required interval. To show this, name an interval around f(χ) and find a value x which relates to the size of the interval around χ, e.g. half of it, that means f(x) falls outside the required interval. If we choose x being half the distance away