Continuity and Differentiability: Continuous Function (For CBSE, ICSE, IAS, NET, NRA 2022)

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Continuity and Differentiability

  • The notion of continuity and differentiability is a pivotal concept in calculus because it directly links and connects limits and derivatives.
  • We have already learned how to prove that a function is continuous, but now we are going to expand upon our knowledge to include the idea of differentiability.
  • In other words, we՚re going to learn how to determine if a function is differentiable.

Continuous Function

  • When in a function, the real value at a point is said to be continuous when at that point, the function of that point is equal to the limit of the function at that point. The continuity exists when the entire domain is continuous.
  • In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, sufficiently small changes in the input of a continuous function result in arbitrarily small changes in its output. If not continuous, a function is said to be discontinuous.
  • The difference, product, and quotient are continuous when it comes to continuous function
  • All the functions which are differential are said to be continuous but the vice versa is not true.

Differentiability

  • Differentiability is when we are able to find the slope of a function at a given point. In other words, a function is differentiable when the slope of the tangent line equals the limit of the function at a given point.
  • What this really means is that in order for a function to be differentiable, it must be continuous and its derivative must be continuous as well.
  • Additionally, we will discover the three instances where a function is not differentiable:
    • Cusp or corner (sharp turn)
    • Discontinuity (jump, point, or infinite)
    • Vertical Tangent (undefined slope)
We Will Discover the Three Instances Where a Function

Logarithmic Differentiation

  • When the differential equation is in the form . Here, the positive values of and is considered.

Rolle՚s Theorem

  • Let us consider,
  • A continuous function
  • R which is continuous on the point differentiable on the point then, and some external point exists such as c in such that .

Mean Value Theorem

Let us consider, a continuous function which is continuous on the point and differentiable on the point , some external point exists such as c in such that

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