AE Do removable discontinuities exist a. Does the fu ncti

Solution 87 AE STEP_BY_STEP SOLUTION Step-1 A continuous function can be formally defined as a f unction f : x y ,where the preimage of every open set in y is open in x. More concretely, a function f(x) in a single variable x is said to be continuous at point x i0, 1. If f(x0) is defined, so that x 0 is in the domain of ‘ f’. 2. lim f(x) exists for x in the domain of f. x x0 3. lim f(x) = f( x ). x x0 0 Left continuous : lixa() = f(a) , then f(x) is called a left continuous at x=a. Right continuous : lim f(x+ = f(a) , then f(x) is called a right continuous at x=a. xa If ,xamf(x) = f(a) = lxa+(x) , then f(x) is called a continuous function at x=a. If , f(x) is not continuous at x =a means , it is discontinuous at x=a. Step-2 Removable discontinuity: A hole in a graph. That is, a discontinuity that can be "repaired" by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can be made connected by filling in a single point. Formally, a removable discontinuity is one at which the limit of the function exists but does not equal the value of