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Formula Sheets
/ Maths
Chapter 16
Continuity and Differentiability
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Continuity
Differentiability
Continuity
f
continuous at
a
⟺
x
→
a
−
lim
f
(
x
)
=
x
→
a
+
lim
f
(
x
)
=
f
(
a
)
sums, products, quotients (denominator
=
0
) and compositions of continuous functions are continuous
Intermediate value theorem
f
continuous on
[
a
,
b
]
,
f
(
a
)
f
(
b
)
<
0
⇒
f
(
c
)
=
0
for some
c
∈
(
a
,
b
)
Differentiability
f
′
(
a
)
=
h
→
0
lim
h
f
(
a
+
h
)
−
f
(
a
)
,
LHD
=
RHD
differentiable
⇒
continuous
;
not conversely (e.g.
∣
x
∣
at
0
)
Common checks
∣
f
(
x
)
∣
may fail where
f
(
x
)
=
0
;
[
x
]
discontinuous at integers
Limits
Differentiation