Question

Let f be a bounded function on [a,b] which is continuous at each point of (a,b). show that f is integrable

Answer 1

f(a) & f(b) are finite.
f is continuous hence,
f(a+h) =f(a) =f(a-h)
f(b+h) =f(b) =f(b-h) for h tending to 0

Integral of f for limits a to b is given by the area bounded by the function

Since function is defined for all f(a), f(a+h),..upto f(b) for h tending to 0

Hence int{ f(x) d(x)} = F(b) - F(a) (for limit a to b)

Therefore, as the answer is finite for finite bounds implies f is integrable