In: Advanced Math
a) Let f(x) be continuous at any point c.
Then there exists a sequence, converging to , such that the sequence converges to .
Then by definition, for , such that for
for and ((By sequential criteria for continuity.))
Now by triangle inequality we know that :
for
Hence for a sequence converging to , the sequence converges to .
Therefore, by sequential criteria for continuity, is continuous.
b) Consider the function
It can easily be proven that this function is not continuous at any point.
But at every point. And is hence, continuous at each and every point.
Therefore, the converse may not be true. That is, if is continuous at a point , that does not mean that will be continuous at .