a function f(x) is said to be continuous at a point x = a of its domain iff lim (x→a) = f(a)
Related concepts
Discontinuity
Removable Discontinuity
Discontinuity of first kind
Discontinuity of second kind
Showing posts with label Continuity and Differentiability. Show all posts
Showing posts with label Continuity and Differentiability. Show all posts
Tuesday, December 2, 2008
Continuity of functions in an interval
A function f(x) is said to be continuous on an open interval (a,b) iff it is continuous at every point on the interval (a,b).
A function f(x) is said to be continuous on a closed interval [a,b] iff
f is continuous at every point on the interval (a,b), i.e., f is continuous on the open interval (a,b) and
lim (x→a+) f(x) = f(a) and lim (x→bˉ) = f(b).
It has to be continuous on (a,b), it has to be continuous at a from right and at b from left.
A function f(x) is said to be continuous on a closed interval [a,b] iff
f is continuous at every point on the interval (a,b), i.e., f is continuous on the open interval (a,b) and
lim (x→a+) f(x) = f(a) and lim (x→bˉ) = f(b).
It has to be continuous on (a,b), it has to be continuous at a from right and at b from left.
Continuous functions
A function f(x) is said to be continuous, if it is continuous at each point of its domain.
Everywhere continous function:
A function f(x) is said to be everywhere continuous, if it is continuous on the entire real line (-∞,∞).
Everywhere continous function:
A function f(x) is said to be everywhere continuous, if it is continuous on the entire real line (-∞,∞).
Cauchy’s definition of continuity
a function f is said to be continuous at a point a of its domain D iff for every ε>0 there exists a δ>0 (dependent on ε) such that
|x-a| <δ => |f(x)-f(a)| < ε
|x-a| <δ => |f(x)-f(a)| < ε
Heine’s definition of continuity
A function f is said to be continuous at a point a of its domain D, if for every sequence an of the points in D converging to a, the sequence of f(an) converges to f(a)
i.e., lim an = a => lim f(an) = f(a).
i.e., lim an = a => lim f(an) = f(a).
21.9 Differentiability at a point
A function f(x) is said to be differntiable or derivable at x+c,
iff lim (x→c) [f(x)-f(c)/(x-c)] exists finitely.
iff lim (x→c) [f(x)-f(c)/(x-c)] exists finitely.
Relation between continuity and differentiability
If a function is differentiable at a point, it is necessarily continuous at that point.
But the converse is not necessarily true.
This means a function may be continuous at a point but may not be differentiable at that point.
But the converse is not necessarily true.
This means a function may be continuous at a point but may not be differentiable at that point.
Differentiability in a set
A function f(x) defined on an openinterval (a,b) is said to be differentiable in open interval (a,b) if it is differentiable at each point of (a,b).
Some results on differentiability
Every polynomial function is differentiable at each x Є R.
The exponential function is differentiable a^x, a>o is differentiable at each x Є R.
Every constant function is differentiable at each x Є R.
The logarithmic function is differentiable at each point in its domain.
The trigonometric and inverse-trigonometrics functions are differentiable at each point in their domains.
The sum, difference, product and quotient of two differentiable functions are differentiable.
The composition of differentiable function is a differential function.
The exponential function is differentiable a^x, a>o is differentiable at each x Є R.
Every constant function is differentiable at each x Є R.
The logarithmic function is differentiable at each point in its domain.
The trigonometric and inverse-trigonometrics functions are differentiable at each point in their domains.
The sum, difference, product and quotient of two differentiable functions are differentiable.
The composition of differentiable function is a differential function.
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