Continuity and Differentiability

Continuity is a function that never ends or does not stop until any exceptions. Differentiability is a point where the slant or an incline tangent line approaches the limitation of the function at a given point. Now, this leads us to some vital implications — all differentiable functions must therefore be continuous, but not all continuous functions are differentiable! Differentiable means the subsidiary exists at each point in its space. Thus, the main way for the subsidiary to exist is on the off chance that the capacity likewise exists (i.e., is constant) on its area. Hence, a differentiable capacity is likewise a persistent capacity. In any case, because a capacity is consistent doesn’t mean its subsidiary (i.e., the slant of the line digression) is characterized wherever in the area.

FORMULAS AND ALGEBRAIC EXPRESSIONS

A differentiable function is a function that can be approximated locally by a direct function. [f(c + h) − f(c) h ] = f (c). The space off is the arrangement of focuses c ∈ (a, b) for which this breaking point exists. If the breaking point exists for each c ∈ (a, b) we say that f is differentiable on (a, b). Continuity can be characterized for a chart y = f(x) as consistent if we can draw the diagram effectively without lifting the pencil at a point. Let f(x) be a genuine esteemed capacity on the subset of genuine numbers and let c be a point existing in the capacity f(x) space. Then, at that point, we say that the capacity f(x) is nonstop at the point x = c then we say that the function f(x) is continuous at the point x = c if we have Limx→cf(x)=f(c)

THEOREMS

If a function is differentiable, it’s likewise Continuous. This property is exceptionally valuable while working with functions, since, supposing that we realize that a function is differentiable, we promptly realize that it’s likewise persistent. Thus, Differentiability implies Continuity.

Theorem By Professor Richard Brown

“A differentiable function automatically becomes a continuous function. After doing several types of research to calculate the secondary of a function at a location where the function was not continuous. Although trying all of this we failed.” This is why Differentiability implies Continuity.
Theorem 0.1 – Function is differentiable in an area or a point this means that it is continuous

Explanation in algebraic terms:

f(x) + g(x) is continuous at the area c = c
f(x) – g(x) is continuous at area x = c
g(x) . g(x) is continuous at area x = c
f(x)/g(x) is continuous at a point x = c, given g(c) ≠ 0

FUNCTIONS OF CONTINUITY AND DIFFERENTIABILITY

The Continuity of a Function can be clarified graphically, or mathematically. In a chart, the progression of a capacity y = f(x) at a point, is a diagram line that goes constantly through the point, with no break. The coherence of a capacity y = f(x) can be noticed logarithmically assuming the worth of the capacity from the left-hand limit is equivalent to the worth of the capacity from the right-hand limit.
Limx→1−1f(x)=Limx→1+1f(x)Limx→1−1f(x)=Limx→1+1f(x)
That is the Value of x = 0.99, 0.998, marginally lesser than 1, has the equivalent f(x) work esteem as that for x = 1.001, 1.0001, which are somewhat more prominent than 1.
The differentiation of a function gives the difference in the function esteem regarding the adjustment of the area of the capacity. The differentiability of a capacity can be perceived both graphically and mathematically. Mathematically the separation of capacity is the slant of the chart of the capacity y = f(x) at the point x=a, in the area of the capacity
Mathematically the separation of the capacity is the adjustment of the worth of the capacity y = f(x) from f(x₁ ) to f(x₂), regarding the adjustment of the space worth of x from x₁ , to x₂ . This can be expressed as dy/dx=f(x₁)−f(x₂)/x₂−x_{1</sub. .}