If f(x) is discontinuous, determine the reason. f(x)= {x^2 +4, x< =1; x+3, x> 1}

for a function to be continuous, it has to be continuous at every number in its domain.for a function  to be continuous at , thenwhich implies that the limit  must exist.======the pieces of the piece wise function are continuous functions. they are polynomials.the only possible place they can be continuous is where the piecewise breaks them up. this is at  .we need to determine if  we need to first find out the value of the limit, if it exists at all.to determine if a limit exists, see if the left- and right-hand limits are equal to each other.for the left-hand limit, use the top piece because that piece is defined  for values that are less than or equal to 1.for the right-hand limit, use the bottom piece because that piece is defined  for values that are greater than 1.the left- and right-hand limits are different.the piecewise function does not converge to a single number at  ; they converge to different values on either side.so f(x) is discontinuous at  because  does not exist, which fails to satisfy the definition of continuity.answer isb) the limit as x approaches 1 does not exist