All About Continuity and Differentiability

A function which is continuous at each point in its domain is known as a continuous function. It has no breaks and no holes in the graph. A function f is said to be continuous at a point c within its domain if it fulfils the following three conditions:

1. There is an existence of f(c) which means function f is defined at c.
2. Limit of f(x) as x approaches c exists. In simple words, it means that both left-hand limit and right hand limit must be equal as x approaches c.
3. The limit of f(x) is equal to f(c) as x approaches c. This can be written mathematically as given-below:

$\underset{ x \to c }{lim} f \left( x \right) = f \left( c \right)$

CBSE board may include definition-based questions to understand if the student knows what this term means. A function f will be differentiable at a point c within its domain if the derivative of f at point c exists. In other words, the function f must have a well-defined tangent line at (c,f(c)). In mathematical terms, the derivative f′(c) is defined as:

$f^{\text{'}} \left( c \right) = \underset{ h \to 0 }{lim} \frac{ f ( c + h ) - f \left( c \right) }{h}$

The NEET exam or JEE Main exam will not ask you direct definition-based questions. Instead, questions around the relation between continuity and differentiability may be asked. Let us try understanding this. 

• If a function is differentiable at a point, it has to be continuous at that point. This happens because the existence of a derivative ensures that there is no jump or break in function at that point.
• A function may be continuous at a point, but it is not compulsory that it will also be differentiable. For instance, function f(x) = ∣x∣ is continuous at x = 0; however, it is not differentiable at that point since it has a cusp there. 
• In short, for a continuous function to be differentiable as well, it must be smooth without any sharp corners.

Logarithmic differentiation is a technique which is used for differentiating functions that are complex products, quotients or powers of other functions. This method takes the natural logarithm of function before differentiating, which makes it a very simple differentiation process. Let us take a look at the steps for logarithmic differentiation since questions based on this will be asked in IIT JAM exam and IISER entrance exam.

1. Start with function y = f(x) that you need to differentiate
2. Take natural logarithm of both sides ln(y) = ln(f(x))
3. Differentiate both the sides of equation with respect to x. Do remember that y is a function of x. Due to this, you will need to use the chain rule on the left side.
4. Derivative of ln(y) with respect to x is $\frac{1}{y} \cdot \frac{dy}{dx}$
5. Derivative of ln (f(x)) with respect to x is $\frac{ f^{\prime } \left(x\right) }{ f \left(x\right) }$
6. Now, once we have differentiated, let us solve for $\frac{dy}{dx}$ by multiplying y on both the sides.

Second Order Derivative

This derivative provides the rate of change of first derivative of a function. In simple terms, second-order derivative measures how rate of change of a function is itself changing. This helps in identifying the points of inflection and determining the nature of critical points. 

Suppose there is a function f(x); the first derivative f′(x) represents the rate of change of f(x) with respect to x. The second-order derivative is denoted as f′′(x) and it is the derivative of first derivative. 

$f^{″} \left(x\right) = \frac{d}{dx} \left( f^{\prime } \right(x\left) \right)$

• Say, f′′(x)>0 on an interval, graph of f(x) is concave up on that interval. This shows that the function has a shape of a cup because it is curving upwards.
• In case f′′(x)<0 on an interval, graph of f(x) is concave down on the interval. The curve in this case, takes the shape of an upside-down cup and it curves downwards.
• Point of inflection occurs where the concavity of function changes i.e. f′′(x) changes sign. At this point, f′′(x) = 0 or it is undefined.

Exponential Functions An exponential function is of the form $f\left(x\right) = a^x$ Here, a > 0 and a ≠ 1.  Most commonly used exponential function is $f\left(x\right) = e^x$ . Here, e is Euler's number which is equal to 2.71828. These functions are continuous throughout which means for any real number x, function $f\left(x\right) = a^x$  is defined without any jumps, breaks or holes in its graph. Exponential functions are differentiable throughout. Derivative of $f\left(x\right) = e^x$ is unique since it is equal to itself. $\frac{d}{dx} e^x = e^x$

Logarithmic Functions This function is of the general form $f\left(x\right) = {\log }_a \left(x\right)$ Here, a>0,  a ≠ 1 and x>0. The most commonly used logarithmic function is natural logarithm, f(x) = ln(x), which is the logarithm to base e. Logarithmic functions are continuous on their own domain (0,∞). They are not defined for x≤0. Therefore, continuity for logarithmic function is not considered outside this interval.  Logarithmic functions are differentiable on their own domain (0,∞). Derivative of a natural logarithm function is: $\frac{d}{dx} \ln \left(x\right) = \frac{1}{x}$ For general logarithmic function $f\left(x\right) = {\log }_a \left(x\right)$ , the derivative will be  $\frac{d}{dx} {\log }_a \left(x\right) = \frac{1}{ x \ln \left(a\right) }$