Give an example of a function, if such function exists (by sketching OR writing it) that satisfies the conditions given in a), b), c) (each separately):
a) f(-2)=4, f is continuous at x=(-2) but has no derivative at this point
b) f is not defined at x=(-3) but has a limit at x=(-3) and is not continuous at x=(-3)
c) f is continuous at x=(-4) from the right and from the left but is not "continuous" at this point
f(x)= |x+2|+ 4 works.
f(x)= 1 for all x except -3 and not defined at x= -3. If you prefer a "formula", or [tex]f(x)= \frac{x^2- 9}{x+ 3} will work.b) f is not defined at x=(-3) but has a limit at x=(-3) and is not continuous at x=(-3)
This can't be done. "continuous at x= -4 from the right" means and "continuous at x= -4 from the left" means . Those both say that f(-4) exists and together they say exists and is equal to f(-4), precisely the condition that f be continuous at x= -4.c) f is continuous at x=(-4) from the right and from the left but is not "continuous" at this point