Differentiability and Continuity is one of the most important topics and it helps students to understand various concepts like continuity at a certain point, derivative of functions, and continuity on a given interval. (image will be updated soon)
Continuity of a function states the characteristics of the function and its functional value. A function is said to be continuous if the curve has no missing points or breaking points in a given interval or domain, that is the curve is continuous at every point in its domain.
A function f(x) is known as a continuous function at a point x = a, in its domain if the following listed three conditions are satisfied-
1. f (a) exists which means that the value of f (a) is finite.
2. Lim x→a f (x) exists, that is the right-hand limit = left-hand limit, and both R.H.S and L.H.S are finite.
Lim x→a f (x) = f (a)
A function f(x) is said to be continuous in the given interval I that is equal to [x1, x2] only if the three conditions listed above are satisfied for each and every point in the given interval I.
A function is said to be continuous in the closed interval [a, b] if
1. f is continuous in (a, b)
2. limx->a+ f(x) = f (a)
3. limx->b- f(x) = f (b)
A function is said to be continuous in the open interval (a, b) if
f (x) is going to be continuous within the unbounded interval (a, b) if at any point within the given interval the function is continuous.
Geometrical Interpretation of Continuity :
Function f is going to be continuous at x = c if there's no break within the graph of the function at the purpose ( c , f(c) ).
In an interval, a function is claimed to be continuous if there's no break within the graph of the function within the entire interval.
Then when can a function be discontinuous?
A function f is discontinuous at x=a if any of the following is true:
1. limx->a+ f(x) and limx->a- f(x) exist but are not equal.
2. limx->a+ f(x) and limx->a- f(x) exists are both equal but not equal to f(a).
3. f (a) is not defined.
In simpler words, if the function is undefined or does not exist, then we say that the function is discontinuous. |
Therefore we arrive at a modified definition of continuity: A function f is continuous at x=a if: limx->a+ f(x) = limx->a- f(x) = f(a)
Note:
Let f and g be two real functions and let c be a point in the common domain of f and g. If the functions f and g are both continuous at x=c then:
1. f+g is continuous at x=c.
2. f-g is continuous at x=c.
3. f•g is continuous at x=c.
4. f/g is continuous at x=c given that g(c) is not zero.
We can show that the given function is continuous at x=4, f(x) = (x² - 2x)/(x - 3) Solution: The given function f(x) is continuous at x=4 because of the following –
f(x) = x(x-2)/(x-3) Nothing is getting canceled, but when we substitute the value of x=4, then the value of the function results to 8.
Since, both the sides R.H.S and L.H.S are equal to 8 when the value of x= 4, the function is continuous. If any of the above conditions are not true, then the equation is said to be discontinuous. |
Function f(x) is said to be differentiable at the point x = an if the derivative of the function f ‘(a) exists at every point in its given domain.
Differentiability formula-
The differentiability formula is defined by -
f’(a) = \[\frac{f(a+h)-f(a)}{h}\]
If a function is continuous at a particular point then a function is said to be differentiable at any point x = a in its domain. The vice versa of this is not always true.
d/dx (sin x) = | cos x |
d/dx (cos x) = | - Sin x |
d/dx (tan x) = | Sec2x |
d/dx (cot x) = | cosec2x |
d/dx (sec x) = | Sec x tan x |
d/dx (cosec x) = | - cosec x cot x |
Here are a few Differentiability and Continuity Problems and solutions!
Question 1) List down the continuity and differentiability formulas.
Solution) The continuity and differentiability formulas are as follows-
The differentiability problems can be solved using the formula-
f’(a) = \[\frac{f(a+h)-f(a)}{h}\]
For a function f to be continuous it should satisfy the three conditions given below-
1. f (a) exists which means that the value of f (a) is finite.
2. Lim x→a f (x) exists, that is the right-hand limit = left-hand limit, and both R.H.S and L.H.S are finite.
3. Lim x→a f (x) = f (a)
Question 2) Explain the continuity of the given function f(x).
Where , f(x ) = sin x . cos x
Solution)We know that cos x and sin x both are continuous functions. We also know that the product of any two continuous functions is also a continuous function.
Therefore, we can say that the function f(x) = sin x . cos x is also a continuous function.
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