Suppose f ( x ¯ ¯ ) ≠ ≠ 0 for some x ¯ ¯ ∈ ∈ ( a , b ) . Since f is continuous, it is nonzero with the same sign for some c , d such that a < c < x ¯ ¯ < d < b . Without loss of generality, assume f ( x ¯ ¯ ) > 0 . Then take an h that is positive on ( c , d ) and zero elsewhere, for example