Question

Show that every differentiable function is continuous (converse is not true i.e., a function may be continuous but not differentiable).

Answer 1

Hint: Here we will use the basic definition of the differential function. Then we will form the condition of a continuous function. We will then show that the function is continuous to prove that every differentiable function is a continuous function.

Complete step-by-step answer:
Let $$f$$ be the differentiable function at $$x=a$$.
Then according to the basic definition of the differentiation, Differentiation of a function is equals to
$$f^{\prime }\left(c\right)=\underset{x\to a}{lim}{\displaystyle \frac{f\left(x\right)-f\left(a\right)}{x-a}}$$……………………….$$\left(1\right)$$
We know that the for a function to be continuous at a point it must satisfy the equation
$$\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$$
We can write the above equation as
$$\Rightarrow \underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)=0$$……………………$$\left(2\right)$$
So, for a function to be continuous it must satisfy the equation $$\left(2\right)$$.
Now we will find the value of $$\underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)$$ for the given differentiable function.
Therefore we can write $$\underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)$$ as,
$$\underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)=\underset{x\to a}{lim}\left({\displaystyle \frac{f\left(x\right)-f\left(a\right)}{x-a}}\left(x-a\right)\right)$$
$$\Rightarrow \underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)=\underset{x\to a}{lim}\left({\displaystyle \frac{f\left(x\right)-f\left(a\right)}{x-a}}\right)\times \underset{x\to a}{lim}\left(x-a\right)$$
By using the equation $$\left(1\right)$$ in the above equation, we get
$$\Rightarrow \underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)=f^{\prime }\left(c\right)\times \underset{x\to a}{lim}\left(x-a\right)$$
By putting the limit on the RHS of the equation, we get
$$\Rightarrow \underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)=f^{\prime }\left(c\right)\times \left(a-a\right)$$
$$\Rightarrow \underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)=f^{\prime }\left(c\right)\times 0$$
$$\Rightarrow \underset{x\to a}{lim}\left(f\left(x\right)-f\left(a\right)\right)=0$$
Hence as per the condition of the equation $$\left(2\right)$$ we can say that the given function $$f$$ is a continuous function.
Hence, every differentiable function is continuous.

Note: Here we have to note that continuous function is the function whose value does not change or value remains constant. When the function is continuous at a point then the left hand limit of the function and the right hand limit of the function is equal to the value of the function at that point.
$$\underset{x\to a^{-}}{lim}f(x)=\underset{x\to a^{+}}{lim}f(x)=f(a)$$
Also, a differentiable function is always continuous but the converse is not true which means a function may be continuous but not always differentiable. A differentiable function may be defined as is a function whose derivative exists at every point in its range of domain.