5 25、证明:函数f:I→R在0∈I处连续←→Vxn∈I,xn→mo(n→0),有limf(xn)=f(xo) 26、设函数∫:I→R在x∈Ⅰ连续,且f(xo)>0。证明:存在xo的一个邻域,在该邻域内 f(x)≥q>0 27、讨论下列函数的连续性,并画出图形 (1)f(x)= (2)f(x) 4 sIn (3)f()=,x≠0 (4)f(x)={] 28、讨论下列函数在指定点处的连续性。若是间断点,说明它的类型 sIn x<0, (1)f(x) (2)f(x) 2x-3,x≠0,x=3 1,x≥0 x=0, 29、讨论下列函数的连续性。若有间断点,说明间断点的类型 (1)f(x) (2)f(x) ,x≠0 1,x≥0; P (q>0,q,p为互质的整数 <1 (3)f(x)= (4)f(x)= 0,x为无理数; x-1x>1 n丌x,x为有理数 x,|x|≤1, (5)f(x) (6)f(x) 0,x为无理数 x|≥ 其中(3)所表示的函数叫黎曼( Riemann)函数 30、试确定常数,使下列函数在x=0处连续: arctan 0, (1)f(x) (2)f(x) 0; 个+x-1,x≠0 0 (3)f(x)= (4)f(x) 1 (5)f()=(1+x),x≠0 ≠0, (6)f(x)5 25  f : I → R # x0 ∈ I @ ⇐⇒ ∀xn ∈ I,xn → x0(n → ∞), $ limn→∞ f(xn) = f(x0). 26 " f : I → R # x0 ∈ I  Æ f(x0) > 0 #8# x0 A9=#e9= f(x) ≥ q > 0 # 27 ^T2/ A:f1oS# (1) f(x) =    x2 − 4 x − 2 , x = 2, 4, x = 2; (2) f(x) =    | sin x x |, x = 0, 1, x = 0; (3) f(x) =    sin x |x| , x = 0, 1, x = 0; (4) f(x)=[x]. 28 ^T2/#p$ @ A#6!Cg qfQ_ (1) f(x) =    x sin 1 x , x< 0, 1, x ≥ 0, x = 0; (2) f(x) =    2 − 1 x − 3 , x = 0, 2, x = 0, x = 3; 29 ^T2/ A#6$Cg qCg Q_ (1) f(x) =    sin x x , x< 0, x2 − 1, x ≥ 0; (2) f(x) =    e − 1 x2 , x = 0, 2, x = 0; (3) f(x) =    1 q , x = p q (q > 0, q, pHhrs), 0, xHE_; (4) f(x) =    cos πx 2 , x ≤ 1, |x − 1|, x> 1; (5) f(x) =    sin πx, xH$_, 0, xHE_; (6) f(x) =    x, |x| ≤ 1, 1, |x| ≥ 1. e' 3 tiujkl Riemann U 30 )5$mM2/# x = 0 @  (1) f(x) =    a + x, x ≤ 0, sin x, x > 0; (2) f(x) =    arctan 1 x , x< 0, a + √x, x ≥ 0; (3) f(x) =    √1 + x − 1 √3 1 + x − 1 , x = 0, c, x = 0; (4) f(x) =    sin x · sin 1 x , x = 0, c, x = 0; (5) f(x) =    (1 + x) 1 x , x = 0, c, x = 0; (6) f(x) =    tan x x , x = 0, c, x = 0