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Mathematics
Problem
Which of the following equalities is necessary and sufficient for a function
f
f
f
to be continuous at
x
=
a
x=a
x
=
a
? a)
lim
x
→
a
−
f
(
x
)
=
lim
x
→
a
+
f
(
x
)
\displaystyle \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)
x
→
a
−
lim
f
(
x
)
=
x
→
a
+
lim
f
(
x
)
b)
lim
x
→
a
+
f
(
x
)
=
f
(
a
)
\displaystyle \lim_{x\to a^+}f(x)=f(a)
x
→
a
+
lim
f
(
x
)
=
f
(
a
)
c)
lim
x
→
a
−
f
(
x
)
=
lim
x
→
a
+
f
(
x
)
=
f
(
a
)
\displaystyle \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)
x
→
a
−
lim
f
(
x
)
=
x
→
a
+
lim
f
(
x
)
=
f
(
a
)
d)
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
=
0
\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}=0
h
→
0
lim
h
f
(
a
+
h
)
−
f
(
a
)
=
0
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