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1) has a removable discontinuity;atx
0
.
2) has an infinite discontinuity at x
0
;
3) has a jump discontinuity at x
0
.
The following statement is often helpful.
Theorem 6
Each elementary function is continuous at every number in its
domain.
Remark 1
There are main elementary functions which include root functions
y = x
a
, a is a real number; exponential functions y = b
x
, b>0,
logarithmic functions y =log
a
x, positive a =1, trigonometric
functions y =sinx, y =cosx, y =tanx, y =cotx or y =ctgx,
y =secx, y =cscx or y =cosecx; and inverse trigonometric
functions y =arcsinx, y = arccos x, y = arctgx, y = arcctg x,
y =arcsecx, y = arccosecx.
As it has been just made clear, the Western notation for some
of trigonometric and inverse trigonometric functions might differ
slightly from the Russian one. Also, one can write sin
−1
x for arcsin x,
etc. This is due to a standard way to denote by f
−1
an inverse
function for f.
y = f(x) is called an elementary function if it can be obtained
from a finite number of main elementary functions (and, possibly,
constants) by applying (a finite number of) operations of addition,
subtraction, multiplication, division, and composition.
We also distinguish between algebraic (they are necessarily elementary
functions) and trancedental (or non-algebraic) functions.
Having made the above reminder (about different classes of functions),
we now return to continuity.
Definition 14
A function f(x) is continuous from the right at x = a if
lim
x→a+
f(x)=f(a), or f(a +0)=f(a)
13
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