diff --git a/src/intervals/interval_operations/boolean.jl b/src/intervals/interval_operations/boolean.jl
index ac46e177b..a4e3392bd 100644
--- a/src/intervals/interval_operations/boolean.jl
+++ b/src/intervals/interval_operations/boolean.jl
@@ -64,7 +64,12 @@ function isinterior(x::Interval, y::Interval)
     return isinterior(bareinterval(x), bareinterval(y))
 end
 
-isinterior(x::AbstractVector, y::AbstractVector) = all(t -> isinterior(t[1], t[2]), zip(x, y))
+function isinterior(x::AbstractVector, y::AbstractVector)
+    n = length(x)
+    m = length(y)
+    n == m || return throw(DimensionMismatch("dimensions must match: x has length $n, y has length $m"))
+    return all(t -> isinterior(t[1], t[2]), zip(x, y))
+end
 
 isinterior(x, y, z, w...) = isinterior(x, y) & isinterior(y, z, w...)
 isinterior(x::Complex, y::Complex) =
@@ -86,7 +91,7 @@ isstrictsubset(x::BareInterval, y::BareInterval) = isinterior(x, y)
 
 isstrictsubset(x::Interval, y::Interval) = isinterior(x, y)
 
-isstrictsubset(x::AbstractVector, y::AbstractVector) = any(t -> isinterior(t[1], t[2]), zip(x, y))
+isstrictsubset(x::AbstractVector, y::AbstractVector) = isinterior(x, y) & any(t -> isinterior(t[1], t[2]), zip(x, y))
 
 isstrictsubset(x, y, z, w...) = isstrictsubset(x, y) & isstrictsubset(y, z, w...)
 isstrictsubset(x::Complex, y::Complex) =
@@ -179,12 +184,13 @@ Test whether `x` is an element of `y`.
 
 Implement the `isMember` function of the IEEE Standard 1788-2015 (Section 10.6.3).
 """
-function in_interval(x::Real, y::BareInterval)
+function in_interval(x::Number, y::BareInterval)
     isinf(x) && return contains_infinity(y)
     return inf(y) ≤ x ≤ sup(y)
 end
+in_interval(x::Complex, y::BareInterval) = in_interval(real(x), y) & iszero(imag(x))
 
-function in_interval(x::Real, y::Interval)
+function in_interval(x::Number, y::Interval)
     isnai(y) && return false
     return in_interval(x, bareinterval(y))
 end
@@ -196,8 +202,8 @@ in_interval(::Interval, ::Interval) =
     throw(ArgumentError("`in_interval` is purposely not supported for two interval arguments. See instead `issubset_interval`"))
 
 in_interval(x::Complex, y::Complex) = in_interval(real(x), real(y)) & in_interval(imag(x), imag(y))
-in_interval(x::Complex, y::Real) = in_interval(real(x), y) & isthinzero(imag(x))
-in_interval(x::Real, y::Complex) = in_interval(x, real(y)) & in_interval(0, imag(y))
+in_interval(x::Complex, y::Number) = in_interval(real(x), y) & iszero(imag(x))
+in_interval(x::Number, y::Complex) = in_interval(x, real(y)) & in_interval(0, imag(y))
 
 in_interval(x) = Base.Fix2(in_interval, x)
 
@@ -328,13 +334,18 @@ isthin(x::Complex) = isthin(real(x)) & isthin(imag(x))
 Test whether `x` contains only `y`.
 """
 isthin(x::BareInterval, y::Number) = inf(x) == sup(x) == y
+isthin(x::BareInterval, y::Complex) = isthin(x, real(y)) & iszero(imag(y))
 
 function isthin(x::Interval, y::Number)
     isnai(x) && return false
     return isthin(bareinterval(x), y)
 end
 
-isthin(x::Complex, y::Number) = isthin(real(x), y) & isthin(imag(x), y)
+isthin(x::Complex, y::Complex) = isthin(real(x), real(y)) & isthin(imag(x), imag(y))
+isthin(x::Complex, y::Number) = isthin(real(x), real(y)) & isthinzero(imag(x))
+isthin(x::Number, y::Complex) = isthin(real(x), real(y)) & iszero(imag(y))
+
+isthin(x::BareInterval, y::Interval) = throw(MethodError(isthin, (x, y)))
 
 """
     isthinzero(x)
@@ -376,4 +387,4 @@ function isthininteger(x::Interval)
     return isthininteger(bareinterval(x))
 end
 
-isthininteger(x::Complex) = isthininteger(real(x)) & isthininteger(imag(x))
+isthininteger(x::Complex) = isthininteger(real(x)) & isthinzero(imag(x))
diff --git a/src/intervals/real_interface.jl b/src/intervals/real_interface.jl
index 676ce3c98..64152fcb5 100644
--- a/src/intervals/real_interface.jl
+++ b/src/intervals/real_interface.jl
@@ -107,9 +107,6 @@ for T ∈ (:BareInterval, :Interval)
         Base.isnan(::$T) =
             throw(ArgumentError("`isnan` is purposely not supported for intervals. See instead `isnai`"))
 
-        Base.isinteger(::$T) =
-            throw(ArgumentError("`isinteger` is purposely not supported for intervals. See instead `isthininteger`"))
-
         Base.intersect(::$T) =
             throw(ArgumentError("`intersect` is purposely not supported for intervals. See instead `intersect_interval`"))
 
@@ -130,3 +127,47 @@ for T ∈ (:BareInterval, :Interval)
             throw(ArgumentError("`setdiff!` is purposely not supported for intervals. See instead `interiordiff`"))
     end
 end
+
+# allow pointwise equality
+
+"""
+    ==(::BareInterval, ::Number)
+    ==(::Number, ::BareInterval)
+    ==(::Interval, ::Number)
+    ==(::Number, ::Interval)
+
+Test whether an interval is the singleton of a given number. In other words, the
+result is true if and only if the interval contains only that number.
+
+!!! note
+    Comparison between intervals is purposely disallowed. Indeed, equality
+    between non-singleton intervals has distinct properties, notably ``x = y``
+    does not imply ``x - y = 0``. See instead [`isequal_interval`](@ref).
+"""
+Base.:(==)(x::BareInterval, y::Number) = inf(x) == sup(x) == y
+Base.:(==)(x::Number, y::BareInterval) = y == x
+function Base.:(==)(x::Interval, y::Number)
+    isnai(x) && return false
+    return bareinterval(x) == y
+end
+Base.:(==)(x::Number, y::Interval) = y == x
+Base.:(==)(x::BareInterval, y::Interval) = throw(MethodError(==, (x, y)))
+Base.:(==)(x::Interval, y::BareInterval) = throw(MethodError(==, (x, y)))
+
+Base.iszero(x::BareInterval) = iszero(inf(x)) & iszero(sup(x))
+function Base.iszero(x::Interval)
+    isnai(x) && return false
+    return iszero(bareinterval(x))
+end
+
+Base.isone(x::BareInterval) = isone(inf(x)) & isone(sup(x))
+function Base.isone(x::Interval)
+    isnai(x) && return false
+    return isone(bareinterval(x))
+end
+
+Base.isinteger(x::BareInterval) = (inf(x) == sup(x)) & isinteger(inf(x))
+function Base.isinteger(x::Interval)
+    isnai(x) && return false
+    return isinteger(bareinterval(x))
+end
diff --git a/test/interval_tests/complex.jl b/test/interval_tests/complex.jl
index 3b2ff5ac7..87ddc206f 100644
--- a/test/interval_tests/complex.jl
+++ b/test/interval_tests/complex.jl
@@ -13,7 +13,7 @@
     @test isequal_interval(a * a, interval(-1))
     @test isequal_interval(a + a, interval(2)*interval(im))
     @test isthin(a - a, 0)
-    @test_broken isthin(a / a, 1)
+    @test isthin(a / a, 1)
 
     @test in_interval(3+2im, c)
     @test isequal_interval(hull(a, b), complex(interval(0, 3), interval(1, 4)))
diff --git a/test/interval_tests/consistency.jl b/test/interval_tests/consistency.jl
index 308817287..c07ba722f 100644
--- a/test/interval_tests/consistency.jl
+++ b/test/interval_tests/consistency.jl
@@ -339,7 +339,10 @@
         @test_throws ArgumentError isempty(x)
         @test_throws ArgumentError isfinite(x)
         @test_throws ArgumentError isnan(x)
-        @test_throws ArgumentError isinteger(x)
+        @test isinteger(x)
+        @test x == 1
+        @test isone(x)
+        @test !iszero(x)
     end
 
 end
diff --git a/test/interval_tests/forwarddiff.jl b/test/interval_tests/forwarddiff.jl
index 21ec86277..7d48f3d0b 100644
--- a/test/interval_tests/forwarddiff.jl
+++ b/test/interval_tests/forwarddiff.jl
@@ -46,7 +46,7 @@ end
         dψ(t)   = ForwardDiff.derivative(ψ, t)
         ddψ(t)  = ForwardDiff.derivative(dψ, t)
         dddψ(t) = ForwardDiff.derivative(ddψ, t)
-        @test ψ′(0)   === dψ(0)   && !isguaranteed(ψ′(0))
+        @test        ψ′(0)   === dψ(0)   && !isguaranteed(ψ′(0))
         @test_broken ψ′′(0)  === ddψ(0)  && !isguaranteed(ψ′′(0)) # rely on `Interval{T}(::Real)` being defined
         @test_broken ψ′′′(0) === dddψ(0) && !isguaranteed(ψ′′′(0)) # rely on `Interval{T}(::Real)` being defined
         t₀ = interval(0)
@@ -62,6 +62,6 @@ end
 
         # g(x) = 2^x # not guaranteed
 
-        @test_broken f′(0) === df(0)
+        @test f′(0) === df(0)
     end
 end