Variables defined within a function can only be referenced within that function.
Therefore, the variable called detector defined in main () cannot be accessed from face_shape_detector_dlib ().
How about modifying it so that it is passed as an argument as follows?

Definition

-def face_shape_detector_dlib (img, detector):
+ def face_shape_detector_dlib (img):

Caller

-face_shape_detector_dlib (img, detector):
+ face_shape_detector_dlib (img):

Additional

import cv2
import dlib
import numpy as np
import imutils
from imutils import face_utils
def face_shape_detector_dlib (img):
    img_rgb = cv2.cvtColor (img, cv2.COLOR_BGR2GRAY)
    # frontal_face_detector class returns rectangle, score, and sub-detector results
    dets, scores, idx = detector.run (img_rgb, 0)
    if len (dets)>0:
        for i, rect in enumerate (dets):
            shape = predictor (img_rgb, rect)
            shape = face_utils.shape_to_np (shape)
            clone = img.copy ()
            cv2.putText (clone, "mouth", (10, 30), cv2.FONT_HERSHEY_SIMPLEX, 0.7, (0, 0, 255), 2)
            # write landmark to image
            for (x, y) in shape [48:68]:
                cv2.circle (clone, (x, y), 1, (0, 0, 255), -1)
            # Get cropped image (ROI) of the part specified by shape
            (x, y, w, h) = cv2.boundingRect (np.array ([shape [48:68]])) #Extract only the mouth part
            roi = img [y: y + h, x: x + w]
            roi = cv2.resize (roi, (100,100))
        return clone, roi
    else:
        return img, None
predictor_path = "./shape_predictor_68_face_landmarks.dat"
predictor = dlib.shape_predictor (predictor_path)detector = dlib.get_frontal_face_detector ()
cap = cv2.VideoCapture (0)
count = 0
while True:
    ret, frame = cap.read ()
    frame = imutils.resize (frame, width = 500)
    frame, roi = face_shape_detector_dlib (frame)
    cv2.imshow ('img', frame)
    if roi is not None:
        cv2.imshow ('roi', roi)
    else:
        cv2.destroyWindow ('roi')
    c = cv2.waitKey (1)
    if c == 27: Press #ESC to close the window
        break
    if c == 32: Save as #space
        count + = 1
        cv2.imwrite ('./ filename% 03.f'% (count) + '. jpg', roi) # 001 ~ Save as sequential number
        print ('save done')
cap.release ()
cv2.destroyAllWindows ()

As the error message says, 'detector' doesn't exist
Let's make it somewhere

import cv2
import dlib
import numpy as np
import imutils
from imutils import face_utils
detector = dlib.get_frontal_face_detector ()
predictor = dlib.shape_predictor ("shape_predictor_68_face_landmarks.dat")
def face_shape_detector_dlib (img, detector):
    img_rgb = cv2.cvtColor (img, cv2.COLOR_BGR2GRAY)
    # frontal_face_detector class returns rectangle, score, and sub-detector results
    dets, scores, idx = detector.run (img_rgb, 0)
    if len (dets)>0:
        for i, rect in enumerate (dets):
            shape = predictor (img_rgb, rect)
            shape = face_utils.shape_to_np (shape)
            clone = img.copy ()cv2.putText (clone, "mouth", (10, 30), cv2.FONT_HERSHEY_SIMPLEX, 0.7, (0, 0, 255), 2)
            # write landmark to image
            for (x, y) in shape [48:68]:
                cv2.circle (clone, (x, y), 1, (0, 0, 255), -1)
            # Get cropped image (ROI) of the part specified by shape
            (x, y, w, h) = cv2.boundingRect (np.array ([shape [48:68]])) #Extract only the mouth part
            roi = img [y: y + h, x: x + w]
            roi = cv2.resize (roi, (80,50))
        return clone, roi
    else:
        return img, None
def main ():
    predictor_path = "./shape_predictor_68_face_landmarks.dat"
    predictor = dlib.shape_predictor (predictor_path)
    detector = dlib.get_frontal_face_detector ()
    cap = cv2.VideoCapture (0)
    count = 0
    while True:
        ret, frame = cap.read ()
        frame = imutils.resize (frame, width = 500)
        frame, roi = face_shape_detector_dlib (frame, detector)
        cv2.imshow ('img', frame)
        if roi is not None:
            cv2.imshow ('roi', roi)
        else:
            cv2.destroyWindow ('roi')
        c = cv2.waitKey (1)
        if c == 27: Press #ESC to close the window
            break
        if c == 32: Save as #space
            count + = 1
            cv2.imwrite ('./ filename% 03.f'% (count) + '. jpg', roi) # 001 ~ Save as sequential number
            print ('save done')
    cap.release ()
    cv2.destroyAllWindows ()
if __name__ == '__main__':
    main ()