# How to prove that a function is continuous

Generally a function is shown continuous by directly taking left hand or right hand limit.But sometimes the same can be shown continuous by letting h tend to zero.What is the difference,would somebody please explain?

• Isn't "letting $h$ tend to zero" taking a limit? What is the distinction you're drawing? – Brian Tung Jun 10 '16 at 18:52
• Type out the two expressions in question – JasonM Jun 10 '16 at 18:54
• Yes.But my puzzle is what is the need of transforming 'x tends to a minus or plus" to "h tends to zero" by taking a-x=h or x-a=h? – Gopesh Jun 10 '16 at 18:57

The definition of continuous function is give as: The function $f$ is continuous at some point $c$ of its domain if the limit of $f(x)$ as $x$ approaches $c$ through the domain of $f$ exists and is equal to $f(c)$. Using the definition is definitely one way to prove that a function is continuous.
I think by "letting h tend to zero" you mean taking the derivative of the function. One property of continuous function is that it has relation with differentiability. Every differentiable function $f:(a,b)\rightarrow R$ is continuous. Although the converse does not hold, we can still use this property to prove that a function is continuous.
There is no difference mathematically; showing $$\lim_{x\to a}f(x)=f(a)$$is exactly the same as showing $$\lim_{h\to0}f(a+h)=f(a).$$Sometimes the formulas that arise with one approach will look nicer or be easier to deal with than the formulas that arise with the other approach.