Final answer:
The integral ∫(a to b) f(x) dx, where function f is continuous on the interval (a, b] and discontinuous at a, is equal to a finite value if the discontinuity is not infinite. option c) is correct option.
Step-by-step explanation:
The question at hand is asking whether a definite integral, ∫(a to b) f(x) dx, where function f is continuous on the interval (a, b] but discontinuous at a, has a specific value. Based on the given information and calculus principles, even if a function f is discontinuous at one endpoint, as long as the function is continuous on the open interval and the discontinuity at the endpoint is not too severe (such as an infinite discontinuity), the integral can still converge to a finite value. Therefore, the integral ∫(a to b) f(x) dx is equal to C) A finite value.
This assumes that the discontinuity is a removable discontinuity or a jump discontinuity, and not an infinite discontinuity which could lead to an infinite result or the integral being undefined.