A function f : A → B is increasing if, for every x and y in A, x ≤ y implies that f(x) ≤ f(y). f is called strictly increasing if, for every x and y in A, x

strictly decreasing function(Noun) Any function of a real variable whose value decreases as the variable increases.

In strictly increasing array A[i] < A[i+1] for 0 <= i < n. Examples: Input : arr[] = { 1, 2, 6, 5, 4} Output : 2 We can change a[2] to any value between 2 and 5. and a[4] to any value greater then 5. Input : arr[] = { 1, 2, 3, 5, 7, 11 } Output : 0 Array is already strictly increasing.

ln x is strictly increasing , since exponential function is strictly increasing.

2 Answers. There is no such function. Suppose that f:R→R is strictly increasing.

Prove that any onto strictly increasing map f:(0,1)→(0,1) is continuous. Since its strictly increasing then for x

Suppose f is a real valued function monotone in the interval [a,b],acan have at most only a countable number of discontinuity in [a,b]. Hence there are uncountably many number of points in [a,b] at which f is continuous. Apply this result to an increasing function.

4. Conclude: A monotone function can only have jump discontinuities. Proof.

The class of monotonic functions consists of both the increasing and decreasing functions. Monotonic functions have no discontinuities of second kind. Let f be monotonic on (a,b).

It is easier to prove that strictly increasing functions form an uncountable set (which implies that the monotone increasing are uncountable as well).

Since left limit not equals right limit not continuous at x =5. Thus there are 3 discontinuities.

To determine this, we find the value of limx→2f(x). The division by zero in the 00 form tells us there is definitely a discontinuity at this point.

Then there are two types of non-removable discontinuities: jump or infinite discontinuities. Removable discontinuities are also known as holes. Jump discontinuities occur when a function has two ends that don’t meet, even if the hole is filled in at one of the ends.

If a function f is continuous at x = a then we must have the following three conditions.

1. f(a) is defined; in other words, a is in the domain of f.
2. The limit. must exist.
3. The two numbers in 1. and 2., f(a) and L, must be equal.

Sal is asked which of the following two functions is continuous on all real numbers: eˣ and/or √x. In general, the common functions are continuous on all the numbers in their domain.

Start by factoring the numerator and denominator of the function. A point of discontinuity occurs when a number is both a zero of the numerator and denominator. Since is a zero for both the numerator and denominator, there is a point of discontinuity there. Since the final function is , and are points of discontinuity.

Functions won’t be continuous where we have things like division by zero or logarithms of zero. Let’s take a quick look at an example of determining where a function is not continuous. Rational functions are continuous everywhere except where we have division by zero.

3 Answers. No, a function can be discontinuous and have a limit. The limit is precisely the continuation that can make it continuous. Let f(x)=1 for x=0,f(x)=0 for x≠0.

Discontinuous functions are functions that are not a continuous curve – there is a hole or jump in the graph. In a removable discontinuity, the point can be redefined to make the function continuous by matching the value at that point with the rest of the function.

In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

(ii) The function y = f (x) is said to be differentiable in the closed interval [a, b] if R f ′ (a) and L f ′ (b) exist and f ′ (x) exists for every point of (a, b).